Course Information

Home
Video Lectures
About The Professor
Official Website

Course

Subject:	Mathematics
Topic:	Multivariable Calculus
Views:	282,079

Educator

Name:	integralCALC Academy (integralCALC)
Type:	Individual

Visit Official Website

Instructor

Name

Krista King

Calculus III: Vector Calculus & Differential Equations

Video Lectures

Displaying all 159 video lectures.

I. Partial Derivatives
Lecture 1 Play Video	Partial Derivatives In this video, Krista King from integralCALC Academy provides an example of a Partial Derivative.
Lecture 2 Play Video	Second Order Partial Derivatives In this video, Krista King from integralCALC Academy explains Second Order Partial Derivatives (Calculus problem example).
Lecture 3 Play Video	Equation of the Tangent Plane in Two Variables In this video, Krista King from integralCALC Academy explains an equation of the tangent plane in two variables (Calculus problem example).
Lecture 4 Play Video	Normal Line to the Surface In this video, Krista King from integralCALC Academy shows how to find the symmetric equations of the normal line to the given surface. In order to do this, you'll need to first find the equation of the tangent plane to the surface. In general, this is the same as finding the equation of the tangent line and normal line to a function, but now we're doing it in 3-dimensional space with multivariable functions.
Lecture 5 Play Video	Linear Approximation in Two Variables In this video, Krista King from integralCALC Academy explains Linear Approximation in Two Variables (Calculus example).
Lecture 6 Play Video	Linearization of a Multivariable Function In this video, Krista King from integralCALC Academy shows how to find the linearization of the multivariable function.
Lecture 7 Play Video	Differential of the Multivariable Function In this video, Krista King from integralCALC Academy shows how to find the differential of a multivariable function.
Lecture 8 Play Video	Chain Rule for Partial Derivatives of Multivariable Functions In this video, Krista King from integralCALC Academy shows how to use chain rule to find partial derivatives of multivariable functions.
Lecture 9 Play Video	Chain Rule and Tree Diagrams of Multivariable Functions In this video, Krista King from integralCALC Academy shows how to use tree diagrams to help you write out the formula for the partial derivative of a multivariable function.
Lecture 10 Play Video	Implicit Differentiation for Partial Derivatives of Multivariable Functions In this video, Krista King from integralCALC Academy shows how to use implicit differentiation to find the partial derivatives of a multivariable function.
Lecture 11 Play Video	Directional Derivatives In this video, Krista King from integralCALC Academy explains directional derivatives (Calculus problem example).
Lecture 12 Play Video	Gradient Vectors In this video, Krista King from integralCALC Academy explains a gradient vector Calculus problem example (or maximum directional derivatives).
Lecture 13 Play Video	Gradient Vectors and the Tangent Plane In this video, Krista King from integralCALC Academy explains a gradient vector and the tangent plane (Calculus problem example).
Lecture 14 Play Video	Gradient Vectors and Maximum Rate of Change In this video, Krista King from integralCALC Academy shows how to use gradient vectors to find maximum rate of change and the direction in which it occurs.
Lecture 15 Play Video	Second Derivative Test: Two Variables In this video, Krista King from integralCALC Academy explains the Second Derivative Test (Calculus problem example).
Lecture 16 Play Video	Local Extrema and Saddle Points of a Multivariable Function In this video, Krista King from integralCALC Academy shows how to use the second derivative test to find local extrema (local maxima and local minima) and saddle points of a multivariable function.
Lecture 17 Play Video	Global Extrema in Two Variables In this video, Krista King from integralCALC Academy explains a global extrema in two variables (Calculus problem example).
Lecture 18 Play Video	Extreme Value Theorem and Extrema in the Set D In this video, Krista King from integralCALC Academy shows how to find global extrema of a multivariable function which is defined for the set of points D, or in the domain D. You'll need to look for critical points in side the set, at the corners of the set, and along the edges of the set. Then you'll need to evaluate those critical points in the original function to identify maxima and minima.
Lecture 19 Play Video	Max Product of Three Real Numbers In this video, Krista King from integralCALC Academy shows how to find three positive numbers with sum 100 and maximum possible product. To do this, write two equations that define the relationship between the numbers, and then use the equations together to reduce the equation of the product to just two variables. Take first-order and second-order partial derivatives. Use the first-order partial derivatives to find critical points of the function, and eliminate all critical points that don't match the criteria of the question. Then use the second-order partial derivatives in the second derivative test to verify that the remaining critical point is a maximum of the product function.
Lecture 20 Play Video	Max Volume of a Rectangular Box Inscribed in a Sphere In this video, Krista King from integralCALC Academy shows how to find the largest possible volume of a rectangular box inscribed in a sphere of radius r. Write down the equation of a sphere in standard form and then write an equation for the volume of the rectangular box. Since the equation for volume is the equation that needs to be optimized, solve the other equation for one of the variables and substitute the value into the volume equation so that the volume equation is reduced to two variables instead of three. Then take first-order partial derivatives of the volume equation, set them equal to 0, and use the resulting equations as a system of simultaneous equations to find critical points. Since we're dealing with a 3D figure in real space, ignore any critical numbers that have negative values for x or y. Plug the resulting critical point into the volume equation in two variables to find a value that represents the maximum volume of the rectangular box.
Lecture 21 Play Video	Points on the Cone Closest to a Point In this video, Krista King from integralCALC Academy shows how to find the points on the surface of a cone which are closest to the given coordinate point in three-dimensional space. If you're looking for the points which are closest together, then you're trying to minimize the distance between the points, which means you need an equation for distance between the points, and this will be the equation you optimize. Write out the distance formula in three variables, and then use the equation of the cone to make a substitution into the distance formula so that the distance formula is only in two variables. Take first-order partial derivatives of the distance formula in order to find a critical point. Then plug the critical point back into the equation of the cone to find the corresponding z-values that give the points on the cone closest to the given coordinate point. Use the second derivative test to verify that the critical point does represent a local minimum of the distance function.
II. Lagrange Multipliers
Lecture 22 Play Video	Lagrange Multipliers (Part I) In this video, Krista King from integralCALC Academy explains Lagrange Multipliers (Calculus example).
Lecture 23 Play Video	Lagrange Multipliers (Part II) In this video, Krista King from integralCALC Academy explains Lagrange Multipliers (Calculus example).
Lecture 24 Play Video	Lagrange Multipliers in Three Dimensions with Two Constraints In this video, Krista King from integralCALC Academy shows how to use Lagrange multipliers to find the extrema of a three-dimensional function, given two constraint functions. In order to complete this problem, you'll need to take partial derivatives of the original function, as well as each of the constraint functions. Then, you'll set up your equations in order to solve for two multipliers and the three variables from the original function. Once you've found values for the original variables, you'll plug those values into the original function. This will give you the value of a maximum or minimum of the function.
III. Double Integrals
Lecture 25 Play Video	Midpoint Rule to Approximate Volume of a Double Integral In this video, Krista King from integralCALC Academy shows how to use midpoint rule to approximate the volume that lies above the given rectangular region defined by the given x-interval by the given y-interval, and below the given function. You'll divide the interval into rectangles and then find the coordinate point that defines the midpoint of each of those rectangles. Then you'll evaluate the original function at each coordinate point, take the sum of those values, and multiply that sum by delta A, which is the area of one of the rectangles.
Lecture 26 Play Video	Riemann Sums to Approximate Volume of a Double Integral In this video, Krista King from integralCALC Academy shows how to use Riemann sums to approximate the volume that lies above the given rectangular region defined by the given x-interval by the given y-interval, and below the given function. You'll divide the interval into rectangles and then find the coordinate points that define the upper right-hand corner of each of those rectangles. Then you'll evaluate the original function at each coordinate point, take the sum of those values, and multiply that sum by delta A, which is the area of one of the rectangles.
Lecture 27 Play Video	Average Value of a Double Integral In this video, Krista King from integralCALC Academy shows how to estimate the average value of a double integral over a rectangle.
Lecture 28 Play Video	Iterated Integrals In this video, Krista King from integralCALC Academy shows how to evaluate iterated integrals, which are just a type of double integrals.
Lecture 29 Play Video	Double Integrals In this video, Krista King from integralCALC Academy shows how to find the double integral of a function, which represents the volume which sits on top of some region, often a rectangular region, but below the given function.
Lecture 30 Play Video	Double Integrals of Type I and Type II Regions In this video, Krista King from integralCALC Academy shows how to use double integrals to find the area enclosed by type I and type II regions.
Lecture 31 Play Video	Double Integrals to Find the Volume of the Solid In this video, Krista King from integralCALC Academy shows how to use double integrals to find the volume of the solid that lies above the given area and under the given function.
Lecture 32 Play Video	Double Integrals to Find Surface Area In this video, Krista King from integralCALC Academy shows how to use double integrals to find the surface area.
Lecture 33 Play Video	Converting Iterated Integrals to Polar Coordinates In this video, Krista King from integralCALC Academy shows how to convert iterated integrals from cartesian coordinates to polar coordinates.
Lecture 34 Play Video	Converting Double Integrals to Polar Coordinates In this video, Krista King from integralCALC Academy shows how to convert double integrals from cartesian coordinates to polar coordinates.
Lecture 35 Play Video	Sketching the Region Given by a Double Polar Integral In this video, Krista King from integralCALC Academy shows how to sketch the region defined by a given double polar integral.
Lecture 36 Play Video	Double Polar Integral to Find Area In this video, Krista King from integralCALC Academy shows how to use a double polar integral to find area.
Lecture 37 Play Video	Double Polar Integral to Find the Volume of the Solid In this video, Krista King from integralCALC Academy shows how to use a double polar integral to find the volume of the solid.
Lecture 38 Play Video	Double Integrals to Find Mass and Center of Mass of the Lamina In this video, Krista King from integralCALC Academy shows how to find mass and center of mass of the lamina using double integrals. Use the double integral formula for mass and the double integral formulas for x bar and y bar to find center of mass.
IV. Triple Integrals
Lecture 39 Play Video	Midpoint Rule for Triple Integrals In this video, Krista King from integralCALC Academy shows how to use the midpoint rule to estimate the volume of a solid. We'll divide the solid cube into eight sub-boxes of equal size and then use the midpoint of each one to estimate volume.
Lecture 40 Play Video	Average Value of the Triple Integral In this video, Krista King from integralCALC Academy shows how to find the average value of a function that lies above a solid. In this problem, the solid is a cube with side length L that lies in the first octant with one vertex at the origin and three sides that lie along the coordinate axes.
Lecture 41 Play Video	Triple Iterated Integrals In this video, Krista King from integralCALC Academy shows how to evaluate a triple iterated integral
Lecture 42 Play Video	Triple Integrals In this video, Krista King from integralCALC Academy shows how to evaluate a triple integral. Find the volume of the region E which lies under the plane and above the region bounded by three curves.
Lecture 43 Play Video	Triple Integrals to Find Volume of the Solid In this video, Krista King from integralCALC Academy shows how to use triple integrals to find the volume of a solid. In this case, we'll find the volume of the tetrahedron enclosed by the three coordinate planes and another function. We'll need to find the limits of integration for each of our three variables, and then decide on an order of integration.
Lecture 44 Play Video	Expressing a Triple Iterated Integral Six Ways In this video, Krista King from integralCALC Academy shows how to change the limits of integration and order of integration to express a triple iterated integral in 6 different ways.
Lecture 45 Play Video	Mass and Center of Mass with Triple Integrals In this video, Krista King from integralCALC Academy shows how to use triple integrals to find mass and center of mass of a solid, E, with a given density function p (rho)
Lecture 46 Play Video	Moments of Inertia with Triple Integrals In this video, Krista King from integralCALC Academy shows how to use triple integrals to find moments of inertia about each of the three coordinate axes.
Lecture 47 Play Video	Cylindrical Coordinates In this video, Krista King from integralCALC Academy shows how to convert cylindrical coordinates to rectangular coordinates.
Lecture 48 Play Video	Converting Triple Integrals to Cylindrical Coordinates In this video, Krista King from integralCALC Academy shows how to convert a triple integral from cartesian coordinates to cylindrical coordinates. You'll need to convert the function itself, the order of integration, and the limits of integration before evaluating the triple integral in cylindrical coordinates.
Lecture 49 Play Video	Volume in Cylindrical Coordinates In this video, Krista King from integralCALC Academy shows how calculate the volume of a solid using a triple integral in cylindrical coordinates. In this problem, the solid lies within a cylinder, above a plane, and below a cone and must be converted from cartesian coordinates to cylindrical coordinates.
Lecture 50 Play Video	Spherical Coordinates In this video, Krista King from integralCALC Academy talks about Spherical Coordinates (Calculus example).
Lecture 51 Play Video	Triple Integral in Spherical Coordinates to Find Volume In this video, Krista King from integralCALC Academy shows how to use a triple integral in spherical coordinates to find the volume of an object, in this case, the ball with center at the origin and radius 5.
Lecture 52 Play Video	Jacobian of the Transformation (2x2) In this video, Krista King from integralCALC Academy shows how to find the Jacobian of the transformation given two equations for x and y, both defined in terms of two other variables, r and theta. Use a 2x2 matrix.
Lecture 53 Play Video	Jacobian of the Transformation (3x3) In this video, Krista King from integralCALC Academy shows how to find the Jacobian of the transformation given three equations for x, y and z, all defined in terms of three other variables, u, v and w. Use a 3x3 matrix.
Lecture 54 Play Video	Plotting Points in Three Dimensions In this video, Krista King from integralCALC Academy shows how to plot points in three dimensional coordinate space (R^3).
V. Vector Calculus & Conic Sections
Lecture 55 Play Video	Distance Formula for Three Variables In this video, Krista King from integralCALC Academy shows how to use the distance formula for three variables in three dimensional coordinate space to find the distance between two points in R^3.
Lecture 56 Play Video	Equation of a Sphere, Plus Center and Radius In this video, Krista King from integralCALC Academy shows how to find the equation of a sphere, put it in standard form, and then find the center and radius of the sphere.
Lecture 57 Play Video	Describing a Region in 3D Space In this video, Krista King from integralCALC Academy shows how to describe a region in three-dimensional coordinate space (R^3).
Lecture 58 Play Video	Using Inequalities to Describe a Region in 3D Space In this video, Krista King from integralCALC Academy shows how to use inequalities to describe a region in three dimensional coordinate space (R^3).
Lecture 59 Play Video	Finding a Vector From Two Points In this video, Krista King from integralCALC Academy shows how to find a vector from two points.
Lecture 60 Play Video	Vector Addition and Combinations of Vectors In this video, Krista King from integralCALC Academy shows how to use the definition of vector addition, also called the triangle law, to find the sum of two vectors, or the combination of two vectors.
Lecture 61 Play Video	Sum of Two Vectors In this video, Krista King from integralCALC Academy talks about the sum of two vectors.
Lecture 62 Play Video	Copying Vectors to Find Combinations of Vectors In this video, Krista King from integralCALC Academy shows how to use the definition of vector addition, also called the triangle law, to find the sum of two vectors, or the combination of two or three vectors.
Lecture 63 Play Video	Unit Vector in the Direction of the Given Vector In this video, Krista King from integralCALC Academy shows how to find a unit vector in the direction of the given vector. Remember that the unit vector is a vector that is 1 unit long, or has a magnitude of 1.
Lecture 64 Play Video	Angle Between a Vector and the x-axis In this video, Krista King from integralCALC Academy shows how to find the angle between a vector and the positive direction of the x-axis.
Lecture 65 Play Video	Magnitude and Angle of the Resultant Force In this video, Krista King from integralCALC Academy shows how to find the magnitude and angle of the resultant force from two vectors.
Lecture 66 Play Video	Dot Product of Two Vectors In this video, Krista King from integralCALC Academy shows how to find the dot product of two vectors. The dot product is also called the scalar product or inner product.
Lecture 67 Play Video	Angle Between Two Vectors In this video, Krista King from integralCALC Academy shows how to find the angle between two vectors in three-dimensional space.
Lecture 68 Play Video	Orthogonal, Parallel or Neither (Vectors) In this video, Krista King from integralCALC Academy shows how to determine whether two vectors are orthogonal to one another, parallel to one another, or neither orthogonal nor parallel.
Lecture 69 Play Video	Acute Angle Between the Lines (Vectors) In this video, Krista King from integralCALC Academy shows how to find the acute angle between two lines using vector representations of both lines. Remember that the acute angle is the angle which is less than 90 degrees. If two lines intersect each other and are anything other than perpendicular, then the intersection creates two equal obtuse angles and two equal acute angles. The acute angle is the one we learn to find in this video.
Lecture 70 Play Video	Acute Angles Between the Curves (Vectors) In this video, Krista King from integralCALC Academy shows how to find the acute angles between two curves by finding their points of intersection, and then the equations of the tangent lines to both curves and the points of intersection. Once you have equations for the tangent lines, you can use the corollary formula for cos(theta) to find the acute angle between the two lines.
Lecture 71 Play Video	Direction Cosines and Direction Angles (Vectors) In this video, Krista King from integralCALC Academy shows how to find the direction cosines and direction angles of a vector. The direction angles are the angles between the vector and the positive directions of the coordinate axes.
Lecture 72 Play Video	Scalar Equation of a Line In this video, Krista King from integralCALC Academy shows how to find the scalar equation of a line with a point and a vector.
Lecture 73 Play Video	Scalar Equation of a Plane In this video, Krista King from integralCALC Academy shows how to find a scalar equation of a plane (Calculus example).
Lecture 74 Play Video	Scalar and Vector Projections In this video, Krista King from integralCALC Academy shows how to find the scalar projections and vector projections of a onto b. The scalar projection is also called the component of b along a.
Lecture 75 Play Video	Cross Product In this video, Krista King from integralCALC Academy shows how to find the cross product of two vectors.
Lecture 76 Play Video	Vector Orthogonal to the Plane In this video, Krista King from integralCALC Academy shows how to find the vector orthogonal to the plane that passes through three points.
Lecture 77 Play Video	Volume of the Parallelepiped Determined by Vectors In this video, Krista King from integralCALC Academy shows how to find the volume of the parallelepiped given three vectors.
Lecture 78 Play Video	Volume of the Parallelepiped with Adjacent Edges In this video, Krista King from integralCALC Academy shows how to find the volume of the parallelepiped given adjacent edges defined by four coordinate points.
Lecture 79 Play Video	Scalar Triple Product to Verify the Vectors are Coplanar In this video, Krista King from integralCALC Academy shows how to use the scalar triple product to verify that the vectors are coplanar.
Lecture 80 Play Video	Vector and Parametric Equations of the Line In this video, Krista King from integralCALC Academy shows how to find the vector equation of the line and the parametric equations of the line.
Lecture 81 Play Video	Parametric and Symmetric Equations of the Line In this video, Krista King from integralCALC Academy shows how to find the parametric equations and symmetric equations of the line.
Lecture 82 Play Video	Symmetric Equations of a Line In this video, Krista King from integralCALC Academy shows how to find the symmetric equations of a line using a point and a vector.
Lecture 83 Play Video	Parallel, Intersecting, Skew and Perpendicular Lines In this video, Krista King from integralCALC Academy shows how to determine whether two lines are parallel, intersecting, skew or perpendicular.
Lecture 84 Play Video	Equation of the Plane Using Vectors In this video, Krista King from integralCALC Academy shows how to find the equation of the plane given three coordinate points in the plane. Use the points to find two vectors that lie in the plane, and then find the cross product of the two vectors, which will be the normal vector to the plane. Write the equation of the plane using one of the coordinate points and the normal vector.
Lecture 85 Play Video	Point of Intersection of a Line and a Plane In this video, Krista King from integralCALC Academy shows how to find the point of intersection between a line defined by parametric equations and a plane. Plug the parametric equations into the equation of the plane so that the equation is defined only in terms of the parameter. Solve for the value of the parameter that defines the point of intersection. Then plug the parameter value into the parametric equations that define the line to get the coordinate point where the line and the plane intersect one another. Verify that this point lies in the plane by plugging the coordinate point into the equation of the plane.
Lecture 86 Play Video	Parallel, Perpendicular, and Angle Between Planes In this video, Krista King from integralCALC Academy shows how to determine whether two planes are parallel or perpendicular, and if the planes are neither parallel nor perpendicular, then to find the angle between the planes. If the planes are parallel, then their normal vectors are parallel. If the planes are perpendicular, then their normal vectors are perpendicular. Therefore, take the coefficients on the x, y and z terms and these are the components of the normal vectors. Set the ratio of the component values equal to each other and if the equation is true, then the normal vectors are parallel and therefore the planes are parallel. Find the dot product of the normal vectors and if the dot product is zero, then the normal vectors are perpendicular so the planes are perpendicular. If the planes are neither parallel nor perpendicular, then use the corollary formula to find the angle between the planes.
Lecture 87 Play Video	Parametric Equations for the Line of Intersection of Two Planes In this video, Krista King from integralCALC Academy shows how to find parametric equations that define the line of intersection of two planes.
Lecture 88 Play Video	Symmetric Equations for the Line of Intersection of Two Planes In this video, Krista King from integralCALC Academy shows how to find symmetric equations for the line of intersection of two planes. In order to do this, we'll find the normal vectors to the planes and take their cross product. We'll find a point on the line of intersection and then use the point and the cross product vector to find the symmetric equations.
Lecture 89 Play Video	Distance Between a Point and a Line (Vectors) In this video, Krista King from integralCALC Academy shows how to use vectors to find the distance between a point and a line, given the coordinate point and parametric equations of the line. Use the parametric equations to find a vector that gives direction numbers and a coordinate point. Find a vector between the two coordinate points. Then take the cross product of the two vectors, and the magnitude of the cross product. Use a distance formula to find the distance between the point and the line.
Lecture 90 Play Video	Distance Between a Point and a Plane (Vectors) In this video, Krista King from integralCALC Academy shows how to find the distance between a point and a plane. Use the distance formula for the component of b along n, where b is the vector between a point in the plane and the given coordinate point, and where n is the normal vector to the plane. The distance formula takes the dot product of the normal vector and the vector b, but simplifies to a formula that simply pulls the components from the coordinate point, and the coefficients from the equation of the plane.
Lecture 91 Play Video	Distance Between Parallel Planes (Vectors) In this video, Krista King from integralCALC Academy shows how to find the distance between the parallel planes using vectors. Verify that the planes are parallel by finding the normal vectors to the planes and proving that the normal vectors are parallel. Then find a coordinate point on one plane, and calculate the distance between that point and the other plane by pulling the components of the normal vectors, and the values from the coordinate point into the distance formula.
Lecture 92 Play Video	Sketching the Quadric Surface In this video, Krista King from integralCALC Academy shows how to sketch a quadric surface and its traces.
Lecture 93 Play Video	Reducing a Quadric Surface Equation to Standard Form In this video, Krista King from integralCALC Academy shows how to simplify an equation into an equation of a quadric surface in standard form.
Lecture 94 Play Video	Domain of the Vector Function In this video, Krista King from integralCALC Academy shows how to find the domain of a vector function when the vector function is given in the form r(t) where r is the vector and t is the parameter that defines each component of the vector.
Lecture 95 Play Video	Limit of the Vector Function In this video, Krista King from integralCALC Academy shows how to find the limit of the vector function.
Lecture 96 Play Video	Sketching the Vector Equation In this video, Krista King from integralCALC Academy shows how to sketch the vector equation. Given the components of the vector, write parametric equations for the vector. Use the parametric equations to find two-variable equations that model the projections of the curve on each of the three coordinate axes. Draw the projections and then draw the curve.
Lecture 97 Play Video	Projections of the Curve Onto the Coordinate Axes In this video, Krista King from integralCALC Academy shows how to sketch the projections of the curve. Projections are like shadows formed by the curve on the three coordinate axes. Given components of the vector equation, you can write parametric equations of the curve. Use the parametric equations to find equations of the projections in terms of only the variables involved in the coordinate axis. Sketch the projections and then use them to draw the real curve.
Lecture 98 Play Video	Vector and Parametric Equations of the Line Segment In this video, Krista King from integralCALC Academy shows how to find the vector equation and parametric equations of the line segment connecting two coordinate points.
Lecture 99 Play Video	Vector Function for the Curve of Intersection of Two Surfaces In this video, Krista King from integralCALC Academy shows how to find the vector function for the curve of intersection of two surfaces, where one surface is a cone and the other surface is a plane.
Lecture 100 Play Video	Derivative of the Vector Function In this video, Krista King from integralCALC Academy shows how to find the derivative of a vector function.
Lecture 101 Play Video	Unit Tangent Vector In this video, Krista King from integralCALC Academy shows how to find the equation of the unit tangent vector to a vector function for a given value of the parameter.
Lecture 102 Play Video	Parametric Equations of the Tangent Line (Vectors) In this video, Krista King from integralCALC Academy shows how to find parametric equations of the tangent line to the vector function at a specific point.
Lecture 103 Play Video	Integral of the Vector Function In this video, Krista King from integralCALC Academy shows how to find the integral of the vector function.
Lecture 104 Play Video	Green's Theorem: One Region In this video, Krista King from integralCALC Academy talks about Green's Theorem in One Region (Calculus problem example).
Lecture 105 Play Video	Green's Theorem: Two Regions In this video, Krista King from integralCALC Academy talks about Green's Theorem in Two Region (Calculus problem example).
VI. First Order Differential Equations
Lecture 106 Play Video	Linear Differential Equations In this video, Krista King from integralCALC Academy talks about Linear Differential Equations (Calculus problem example).
Lecture 107 Play Video	Circuits and Linear Differential Equations In this video, Krista King from integralCALC Academy shows how to use linear differential equations to solve basic problems of electric circuits.
Lecture 108 Play Video	Linear Differential Equation Initial Value Problem In this video, Krista King from integralCALC Academy shows how to solve a linear differential equations initial value problem.
Lecture 109 Play Video	Differential Equations In this video, Krista King from integralCALC Academy talks about Differential Equations.
Lecture 110 Play Video	Change of Variable to Solve a Differential Equations In this video, Krista King from integralCALC Academy shows how to use a change of variable to solve a separable differential equation.
Lecture 111 Play Video	Separable Differential Equations Initial Value Problem In this video, Krista King from integralCALC Academy shows how to solve a separable differential equations initial value problem.
Lecture 112 Play Video	Mixing Problems with Separable Differential Equations In this video, Krista King from integralCALC Academy shows how to solve mixing problems using separable differential equations
Lecture 113 Play Video	Euler's Method (Part I) In this video, Krista King from integralCALC Academy talks about Euler's Method (Calculus example).
Lecture 114 Play Video	Euler's Method (Part II) In this video, Krista King from integralCALC Academy talks about Euler's Method (Calculus example).
Lecture 115 Play Video	Euler's Method (Part III) In this video, Krista King from integralCALC Academy talks about Euler's Method (Calculus example).
Lecture 116 Play Video	Sketching Direction Fields In this video, Krista King from integralCALC Academy shows how to sketch a direction field using the equation for the first derivative in terms of x and y, then learn how to sketch a solution in the direction field that passes through a given point.
Lecture 117 Play Video	Population Growth In this video, Krista King from integralCALC Academy talks about Population Growth (Calculus problem example).
Lecture 118 Play Video	Logistic Growth Model of a Population In this video, Krista King from integralCALC Academy shows how to write a logistic growth equation that models the population over time given the initial population, the carrying capacity, and the population after one year. Then learn how to use the logistic growth equation to find the population at a given time.
Lecture 119 Play Video	Predator-Prey Systems In this video, Krista King from integralCALC Academy shows how to determine which variable represents the predator population, and which represents the prey population, how to determine if the predator or prey populations are effected by any other food source than the other species (carrying capacity), how to determine whether the system is one of cooperation, competition, or predator-prey, and how to solve the system of differential equations for equilibrium solutions, and then interpret those equilibrium solutions.
VII. Second Order Differential Equations
Lecture 120 Play Video	Second-Order Differential Equations In this video, Krista King from integralCALC Academy shows how to find second-order differential equations.
Lecture 121 Play Video	Equal Real Roots of Second-Order Homogeneous Differential Equations In this video, Krista King from integralCALC Academy shows how to find the general solution of a second-order homogeneous differential equation when the equation gives equal real roots.
Lecture 122 Play Video	Complex Conjugate Roots of Second-Order Homogeneous Differential Equations In this video, Krista King from integralCALC Academy shows how to find the general solution of a second-order homogeneous differential equation when the equation gives complex conjugate roots.
Lecture 123 Play Video	Second-Order Differential Equations: Initial Value Problems (Example 1) In this video, Krista King from integralCALC Academy shows how to solve second-order differential equations with distinct real roots.
Lecture 124 Play Video	Boundary Value Problem, Second-Order Homogeneous Differential Equation, and Distinct Real Roots In this video, Krista King from integralCALC Academy shows how to solve a boundary value problem given a second-order homogeneous differential equation and two initial conditions. In this particular problem, the roots of the equation are distinct real roots and the general solution to the differential equation is written with r_1 and r_2.
Lecture 125 Play Video	Boundary Value Problem, Second-Order Homogeneous Differential Equation, and Complex Conjugate Roots In this video, Krista King from integralCALC Academy shows how to solve a boundary value problem given a second-order homogeneous differential equation and two initial conditions. In this particular problem, the roots of the equation are complex conjugate roots and the general solution to the differential equation is written with alpha and beta.
Lecture 126 Play Video	Second-Order Differential Equations: Working Backwards In this video, Krista King from integralCALC Academy shows how to work backwards in Second-Order Differential Equations (Calculus problem example).
Lecture 127 Play Video	Second-Order Non-Homogeneous Differential In this video, Krista King from integralCALC Academy talks about Second-Order Non-Homogeneous Differential Equations (Calculus problem example).
Lecture 128 Play Video	Variation of Parameters for Differential Equations In this video, Krista King from integralCALC Academy shows how to use variation of parameters to find the particular solution of a non-homogeneous second-order differential equation.
Lecture 129 Play Video	Second-Order Non-Homogeneous Differential Equations: Initial Value Problem In this video, Krista King from integralCALC Academy talks about Second-Order Non-Homogeneous Differential Equation Initial Value Problem.
VIII. Laplace Transforms
Lecture 130 Play Video	Laplace Transforms Using the Definition In this video, Krista King from integralCALC Academy talks about Laplace Transforms using the definition (Calculus problem example).
Lecture 131 Play Video	Laplace Transforms Using a Table In this video, Krista King from integralCALC Academy talks about Laplace Transforms using a table (Calculus problem example).
Lecture 132 Play Video	Initial Value Problems with Laplace Transforms In this video, Krista King from integralCALC Academy talks about Initial Value Problems with Laplace Transforms (Calculus problem example).
Lecture 133 Play Video	Laplace Transforms and Integration by Parts with Three Functions In this video, Krista King from integralCALC Academy talks about Laplace Transforms and Integration by Parts with Three Functions
Lecture 134 Play Video	Inverse Laplace Transform In this video, Krista King from integralCALC Academy talks about Inverse Laplace Transform (Calculus problem example).
Lecture 135 Play Video	Convolution Integral for Initial Value Problems In this video, Krista King from integralCALC Academy shows how to use the convolution integral to calculate the inverse laplace transform of a second-order non-homogeneous differential equation initial value problem.
Lecture 136 Play Video	Exact Differential Equations In this video, Krista King from integralCALC Academy shows how to solve exact differential equations.
IX. Lagrange Multipliers
Lecture 137 Play Video	Lagrange Multipliers and Three Dimensions, One Constraint In this video, Krista King from integralCALC Academy shows how to solve a lagrange multiplier problem with three variables (three dimensions) and only one constraint equation. We'll find the extrema of the function subject to the constraint.
Lecture 138 Play Video	Limit of the Multivariable Function In this video, Krista King from integralCALC Academy shows how to find the limit of the multivariable function. We'll test the limit as we approach the point along different paths. If we get different values along any two paths, then we'll be able to state that the limit does not exist (DNE). Otherwise, if the value appears to be the same along every path that we test, then we'll have to use the precise definition of the limit for multivariable functions (epsilon-delta definition) in order to prove that the limit exists.
Lecture 139 Play Video	Minimum Distance Between the Point and the Plane In this video, Krista King from integralCALC Academy shows how to find the minimum distance between a point and a plane in three-dimensional coordinate space. This will be an application of optimization of multivariable functions in which we optimize the distance formula.
Lecture 140 Play Video	Precise Definition of the Limit for Multivariable Functions In this video, Krista King from integralCALC Academy talks about the precise definition of the limit for multivariable functions, also known as the epsilon-delta definition of the limit.
Lecture 141 Play Video	Critical Points of Multivariable Functions In this video, Krista King from integralCALC Academy shows how to find the critical points (the points where the function changes direction) of a multivariable function. In order to find critical points, we'll need to take partial derivatives of the function with respect to both variables, and then set each of them equal to 0. We'll use the equations together as a system of linear equations (simultaneous equations) to solve for the unique solution, which will be the critical point.
Lecture 142 Play Video	Discontinuities of a Multivariable Function In this video, Krista King from integralCALC Academy uses multiple examples, showing how to find the discontinuities in a multivariable function.
Lecture 143 Play Video	Domain of a Multivariable Function In this video, Krista King from integralCALC Academy shows how to find the domain of a multivariable function, specifically the domain of a multivariable logarithmic function.
Lecture 144 Play Video	Arc Length of a Vector Function In this video, Krista King from integralCALC Academy shows how to find the arc length of a vector function when the vector function is given in terms of a parameter value t, where t is defined over a closed interval.
Lecture 145 Play Video	Area of the Surface In this video, Krista King from integralCALC Academy shows how to find the area of a surface, given the equation of the surface and the boundaries of the surface defined as the equation of a cylinder.
Lecture 146 Play Video	Tangential and Normal Components of the Acceleration Vector In this video, Krista King from integralCALC Academy shows how to find the tangential and normal components of an acceleration vector when the acceleration vector is given as a vector function.
X. Line Integrals
Lecture 147 Play Video	Curl and Divergence In this video, Krista King from integralCALC Academy shows how to find the curl and divergence of a vector field using the formulas for curl and divergence.
Lecture 148 Play Video	Curvature of the Vector Function In this video, Krista King from integralCALC Academy shows how to find the curvature of a vector function using the formula for curvature.
Lecture 149 Play Video	Independence of Path In this video, Krista King from integralCALC Academy shows how to prove that a line integral is independent of path. We'll do this by showing that the vector field is conservative, knowing that any line integral inside a conservative vector field will be independent of path.
Lecture 150 Play Video	Line Integral of a Curve In this video, Krista King from integralCALC Academy shows how to find the line integral of a curve, when we're looking for the line integral under a function and over a line segment from one coordinate point to another coordinate point.
Lecture 151 Play Video	Line Integral of a Vector Function In this video, Krista King from integralCALC Academy shows how to find the line integral of a vector function, when the vector function is given in three variables (three dimensions).
Lecture 152 Play Video	Maximum Curvature of the Function In this video, Krista King from integralCALC Academy shows how to find the maximum curvature of the function.
Lecture 153 Play Video	Normal and Osculating Planes In this video, Krista King from integralCALC Academy shows how to find the equations of the normal and osculating planes of a parametric equation.
Lecture 154 Play Video	Parametric Representation of the Surface In this video, Krista King from integralCALC Academy shows how to find the parametric representation of the surface, specifically the parametric representation of the part of a sphere that lies above a cone.
Lecture 155 Play Video	Points on the Surface In this video, Krista King from integralCALC Academy shows how to figure out whether or not the given coordinate points lie on the given surface, when the surface is given as a parametric vector function in terms of two parameters.
Lecture 156 Play Video	Potential Function of a Conservative Vector Field In this video, Krista King from integralCALC Academy shows how to find the potential function of a conservative vector field.
Lecture 157 Play Video	Potential Function of the Conservative Vector Field to Evaluate a Line Integral In this video, Krista King from integralCALC Academy shows how to use the potential function of a conservative vector field to evaluate a line integral when we're looking for the line integral from one coordinate point to another coordinate point.
Lecture 158 Play Video	Potential Function of the Conservative Vector Field, Three Dimensions In this video, Krista King from integralCALC Academy shows how to find the potential function of a conservative vector field when the conservative vector field is given in three variables (three dimensions).
Lecture 159 Play Video	Re-parametrizing the Curve in Terms of Arc Length In this video, Krista King from integralCALC Academy shows how to reparametrize the curve in terms of arc length, from t=0 in the direction of increasing t.