Course Information

Course

Subject:	Mathematics
Topic:	Multivariable Calculus
Views:	78,985

Educator

Name:	Math Doctor Bob (Robert Donley)
Type:	Individual

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Calculus with Dr. Bob VII: Multivariable Calculus

Video Lectures

Displaying all 67 video lectures.

Lecture 1 Play Video	Equation of a Parabola 1 Multivariable Calculus: Find the vertex, focus, and directrix of the parabola 72y + 12x^2 - 48x + 192 = 0. Then sketch the graph.
Lecture 2 Play Video	Equation of a Parabola 2 Multivariable Calculus: A parabola has directrix x = -2 and focus at (10, 4). Find the equation of the parabola.
Lecture 3 Play Video	Equation of an Ellipse 1 Multivariable Calculus: Find the center, vertices, foci, and eccentricity of the ellipse 4x^2 - 8x + 9y^2 + 36y + 4 = 0. Sketch the ellipse.
Lecture 4 Play Video	Equation of an Ellipse 2 Multivariable calculus: An ellipse has eccentricity e = 12/13 and vertices at (3, 2) and (3, 28). Find the foci of the ellipse and determine the length of string needed to trace out the ellipse from its foci.
Lecture 5 Play Video	Equation of a Hyperbola 1 Multivariable Calculus: Find the center, vertices, foci, and asymptotes of the hyperbola 4x^2 - 16x - y^2 + 6y - 9 = 0. Then sketch the hyperbola.
Lecture 6 Play Video	Equation of Hyperbola 2 Multivariable Calculus: A hyperbola has asymptotes y = 3/2 x + 4 and y = -3/2 x - 2, and one vertex at (-2, 4). Find the center, foci, and eccentricity of the hyperbola. Sketch.
Lecture 7 Play Video	Example of Equation of a Sphere Multivariable Calculus: A sphere has center (0,3,4) and passes through the point (1,1,1). Find the points on the sphere above the point (sqrt(6), 1) in the xy-plane.
Lecture 8 Play Video	Equation of a Sphere Given Diameter Multivariable Calculus: Find the equation of a sphere if endpoints of one diameter are given as (6,1,0) and (0,1,8).
Lecture 9 Play Video	Equation of Sphere Given Tangent Plane 1 Multivariable Calculus: Find the equation of the sphere with center (-4, 2, 3) and tangent to the plane 2x-y-2z + 7 = 0. In Part 1, we use vector analysis to find the radius. (In Part 2, we minimize the distance function by finding a critical point; in Part 3, we use Lagrange multipliers.)
Lecture 10 Play Video	Equation of Sphere Given Tangent Plane 2 Multivariable Calculus: Find the equation of the sphere with center (-4,2,3) and tangent to the plane 2x-y-2z=-7. In this part, we find the minimum of the distance function using partial derivatives.
Lecture 11 Play Video	Equation of Sphere Given Tangent Plane 3 Multivariable Calculus: Find the equation of the sphere with center (-4,2,3) and tangent to the plane 2x-y-2z=-7. In this part, we use Lagrange multipliers to find the point of contact and the radius.
Lecture 12 Play Video	Angle Between Two Vectors Using Dot Product Multivariable Calculus: Find the angle between the vector u=(sqrt(3)+1, sqrt(3)-1) and v=(sqrt(3), -1) using the dot product.
Lecture 13 Play Video	Vector Decomposition of (2,2,1) Along (1,1,1) Multivariable Calculus: Decompose the vector u = (2,2,1) as sum of vectors parallel and perpendicular to v = (1,1,1). The key step is to derive the formula for the parallel direction: u_par = proj_v(u) = (u dot v)/\|\|v\|\|^2 v.
Lecture 14 Play Video	Unit Vector Perpendicular to Two Vectors Multivariable Calculus: Find a unit vector perpendicular to the vectors u = (1,2,1) and v = (2,1,1). The main tool is the cross product.
Lecture 15 Play Video	Area of Parallelogram in Three Space Multivariable Calculus: Using the cross product, find the area of the parallelogram with corners at the points (1,0,1), (2,1,3), (3,05), and (4,1,7).
Lecture 16 Play Video	Volume of a Parallelepiped Multivariable Calculus: Find the volume of the parallelepiped based at the origin with adjacent sides as position vectors (1,2,3), (1,0,2), and (0,5,6). This provides an application of the triple product.
Lecture 17 Play Video	Diagonal Lengths of a Parallelepiped Multivariable Calculus: Consider the parallelepiped in R^3 based at the origin with adjacent edges given by the vectors u = (1,1,-1), v=(1,2,2) and w=(2,2,0). Find the lengths of the 4 space diagonals.
Lecture 18 Play Video	Example of Symmetric Equations of a Line Multivariable Calculus: Find the symmetric equations of the line through the point (1,0,3) and perpendicular to the plane x+2y-z=6.
Lecture 19 Play Video	Equation of a Parallel Line Multivariable Calculus: Find the parametric equations of the line through the point (-1,3,2) and parallel to the line defined by the symmetric equations (x-2)/3 = (y-3)/-1 = (z-4)/5.
Lecture 20 Play Video	Example of Intersecting Lines Multivariable Calculus: Consider the line L1 with symmetric equations (x-1)/2 = (z+1)/-1, y=2. Determine if the following lines are parallel, equal, intersecting, or skew. (a) (x-5)/2 = (y-2)/1 = (z+3)/-1, (b) (z-1)/2 = (y-1)/1 = z/-1, and (c) x(s) = -19 + 10s, y(s) = 2, z(s) =9 -5s.
Lecture 21 Play Video	Angle Between Two Planes Multivariable Calculus: Consider the planes x-y+z = 4 and 2x + y + z = 10. Determine whether the planes are parallel, perpendicular, or neither. If neither, find the cosine of the angle between them.
Lecture 22 Play Video	Planes: Parallel, Equal, or Intersecting? Multivariable Calculus: Determine whether the following planes are equal, parallel, or intersecting with the plane x -2y + z = 7: (a) x+y+z=2, (b) -2x+4y - 2z = -14, and (c) -2x +4y -2z = 0.
Lecture 23 Play Video	Line of Intersection of Two Planes Multivariable Calculus: Are the planes 2x - 3y + z = 4 and x - y +z = 1 parallel? If not, find the equation of the line of intersection in parametric and symmetric form. The cross product is used to find the direction of the line.
Lecture 24 Play Video	Equation of a Plane Containing a Point and a Line Multivariable Calculus: Find the equation of the plane containing the origin and the line r(t) = (2+3t, 4+t, 1+t). The key step involves using the cross product to find a normal vector.
Lecture 25 Play Video	Equation of a Plane Through Three Points Multivariable Calculus: Find the equation of the plane through the three points (1,1,-1), (1,0,1), and (2,2,0).
Lecture 26 Play Video	Example of Plane-Line Intersections Multivariable Calculus: Consider the plane x - 2y + z = 7. Are the following lines parallel to the plane? Find all points of intersection. (a) (x-2)/3 = y/-1 = (z-2/)1 (b) (x-2)/1 = y/1 = (z+2)/1, and (c) (x-4)/1 = (y-1)/1 = (z-5)/1.
Lecture 27 Play Video	Domain of a Vector-Valued Function Multivariable Calculus: Find the domain of the vector-valued function r(t) = (ln(t), (t-1)/(t^2-1), sqrt(t+1) ).
Lecture 28 Play Video	Limit and Derivative of Vector Function Multivariable Calculus: Consider the vector function r(t) = ( (t-4)/(sqrt(t)-2), t+6, e^{t-4}). Find lim(4) r(t) and r'(9).
Lecture 29 Play Video	Example of Position, Velocity and Acceleration in Three Space Multivariable Calculus: A particle moves through space with acceleration function a(t) = (1, t, 2). At time t=0, the position is (0, 2, 4) and the velocity is (3, 0, 1). Find the position function r(t) and the velocity function v(t).
Lecture 30 Play Video	Tangent Line to a Parametrized Curve Multivariable Calculus: Find the parametric and symmetric equations of the tangent line to the curve r(t) = (cos(t), sin(t), t) when t = pi/2. Where does the line intersect the xy-plane?
Lecture 31 Play Video	Angle of Intersection Between Two Curves Multivariable Calculus: Find the angle of intersection between the curves r1(t) = (1+t, t, t^3) and r2(t) = (cos(t), sin(t), t^2) at the point (1, 0, 0).
Lecture 32 Play Video	Unit Tangent and Normal Vectors for a Helix Multivariable Calculus: Find the unit tangent vector T(t), unit normal vector N(t), and curvature k(t) of the helix in three space r(t) = (3sint(t), 3cos(t), 4t). We also calculate the unit binormal vector B(t).
Lecture 33 Play Video	Sketch/Area of Polar Curve r = sin(3O) Multivariable Calculus: Sketch the polar curve r(O) = sin(3O) as O goes from 0 to pi. Then find the area between the origin and the curve from O = 0 to O = pi/3.
Lecture 34 $Arc Length along Polar Curve r = e^{-O}$ Play Video	Arc Length along Polar Curve r = e^{-O} Multivariable Calculus: Find the arc length along the polar curve r(O) = e^{-O} from O = 0 to O = ln(2).
Lecture 35 Play Video	Showing a Limit Does Not Exist Multivariable Calculus: Show that the limit(0,0) xy/(x^2 + y^2) does not exist by finding two curves in the xy-plane along which the limit does not agree. We use the curves y=x and y=2x. Failure of the limit is also explained using polar coordinates.
Lecture 36 Play Video	Contour Map of f(x,y) = 1/(x^2 + y^2) Multivariable Calculus: Sketch the contour map of f(x,y) = 1/(x^2 + y^2). Label the level curves at c= 0, 1, 1/4, 4. We sketch the graph in three space to check our map.
Lecture 37 Play Video	Sketch of an Ellipsoid Multivariable Calculus: Sketch the surface x^2/9 + y^2/16 + z^2/4 = 1. Show the sections in the xy-, xz-, and yz-planes. For more videos like these, please visit the Multivariable Calculus playlist at my channel.
Lecture 38 Play Video	Sketch of a One-Sheeted Hyperboloid Multivariable Calculus: Sketch the one-sheeted hyperboloid x^2 + y^2/4 - z^2/9 =1. Show the traces in the xy-, xz-, and yz-planes.
Lecture 39 Play Video	Sketch of a Double-Napped Cone Multivariable Calculus: Sketch the double-napped cone x^2 - y^2 + z^2 = 0. Show the traces in the xy-, xz-, and yz-planes.
Lecture 40 Play Video	Example of Implicit Differentiation with Several Variables Multivariable Calculus: Find all first partial derivatives of w in the equation x^2y + y^2 + zw + w^2 = 2 at the point (x,y,z,w) = (1,0,-1,2).
Lecture 41 Play Video	Gradient of f(x,y) = yx^2 + cos(xy) Multivariable Calculus: Find the directions of maximal increase and decrease of the function f(x,y) = x^2y + cos(xy).
Lecture 42 Play Video	Tangent Plane to x^2 - xy - y^2 -z = 0 Multivariable Calculus: Find the equation of the tangent plane to the surface x^2 - xy - y^2 -z = 0 at the point (2,1,1).
Lecture 43 Play Video	Lagrange Multiplier: Single Constraint Multivariable Calculus: A rectangular box has one corner at the origin and another on the plane x + y + 2z = 1. Use Lagrange multipliers to find the dimensions that maximize volume.
Lecture 44 Play Video	Optimization on Ellipse in R^3 1: Parametrization Method Multivariable Calculus: Consider the ellipse given as the intersection of the cylinder x^2 + y^2 =1 and the plane y = z + 1 in three-space. Using a parametrization, we find the maximum and minimum values of f(x,y,z) = xz on the ellipse. In Part 2, same problem with Lagrange Multipliers with two constraints.
Lecture 45 Play Video	Optimization on Ellipse in R^3 2: Lagrange Multipliers with Two Constraints Multivariable Calculus: We find the maximum and minimum values of the function f(x,y,z) = xz on the intersection of the cylinder x^2 + y^2 = 1 and the plane z=y-1 in R^3. In this part, we use the method of Lagrange multipliers with two constrains.
Lecture 46 Play Video	Example of Chain Rule for Partial Derivatives Multivariable Calculus: Find del w/del s and del w/del t if w(x, y) = x^2 - y^2, x(s,t) = cos(s+3t) and y(s,t) = sin(s+3t). We verify the answer by using double angle formulae for sine and cosine.
Lecture 47 Play Video	Second Partials Test for f(x,y) = x^3 + 3xy + y^3 Multivariable Calculus: Find all local maxima/minima and saddle points for the function f(x,y) = x^3 + 3xy + y^3. We apply the Second Partials Test.
Lecture 48 Play Video	Directional Derivative of f(x,y,z) = xy + yz Multivariable Calculus: Find the directional derivative of the function f(x,y,z) = xy + yz in the direction 2i - 2j + k at the point (1,2,4).
Lecture 49 Play Video	Linear Approximation to f(x,y) = x^2y^2 + x Multivariable Calculus: Find the linear approximation to the function f(x, y) = x^2 y^2 + x at the point (2, 3). Then approximate (2.1)^2 (2.9)^2 + 2.1.
Lecture 50 Play Video	Taylor Polynomial of f(x,y) = ycos(x+y) Multivariable Calculus: Find the cubic approximation to f(x,y) = ycos(x+y). Then approximate (.1) cos(.2). The polynomial can be checked using the Maclaurin series for cos(x).
Lecture 51 Play Video	Conversion From Rectangular Coordinates Multivariable Calculus: Convert the rectangular point (x,y,z) = (-1/2,-1/2,0) into cylindrical and spherical coordinates.
Lecture 52 Play Video	Conversion From Cylindrical Coordinates Multivariable Calculus: Convert the cylindrical point (r, theta, z) = (2, 5pi/3, -2) into rectangular and spherical coordinates.
Lecture 53 Play Video	Conversion from Spherical Coordinates Multivariable Calculus: Suppose we have the spherical point (rho, theta, phi) = (2, 3pi/4, pi/3). Find the rectangular and cylindrical coordinates of the point.
Lecture 54 Play Video	Examples of Double and Triple Integrals Multivariable Calculus: Calculate the following iterated integrals: (a) int_0^1 int_0^{y^2} e^{y^3} dx dy, and (b) int_0^1 int_0^{x^2} int_{0}_y zy^2 dz dy dx.
Lecture 55 Play Video	Center of Mass for a Rectangle of Variable Density Multivariable Calculus: Find the center of mass of the rectangle in the plane (x from 0 to 1, y from 0 to 2) with density function rho(x,y) = xy.
Lecture 56 Play Video	Interchange of Limits of Integration Multivariable Calculus: Compute the integral int_0^1 int_{x=y^2}^1 ysin(x^2) dx dy by interchanging the order of integration.
Lecture 57 Play Video	Integral in Polar Coordinates Multivariable Calculus: Compute the integral of y/sqrt(x^2+y^2) over the first quadrant region bounded by the circles of radius 1 and 3.
Lecture 58 Play Video	Area Between Polar Curves r = 2/cos(θ) and r = 4cos(θ) Multivariable Calculus: Find the area between the polar curves r = 2/cos(θ) and r = 4cos(θ) where x is greater than 2. We use the integration formula for regions defined by polar coordinates in the plane and verify our answer by using a familiar formula.
Lecture 59 Play Video	Integral of exp(-x^2) (HD Version) Multivariable calculus: We calculate the integral of exp(-x^2) over the real line. This calculation involves a double integral, from which values are calculated using polar coordinates. Added: a numerical estimate, and an added tighter proof with the Squeeze Theorem.
Lecture 60 Play Video	Surface area of z = (x^2+y2)^1/2 Multivariable Calculus: Find the area of the surface z = (x^2 + y^2)^1/2 over the unit disk in the xy-plane. After computing, we re-derive the area formula.
Lecture 61 Play Video	Mass of Solid as a Triple integral in Rectangular Coordinates Multivariable Calculus: Consider the solid E = {y: 0 to 1, x: 0 to y, z:0 to x + y}. If the density function on E is d(x,y,z) = yz, use a triple integral to calculate the mass of E.
Lecture 62 Play Video	Volume of Truncated Paraboloid in Cylindrical Coordinates Multivariable Calculus: Using a triple integral, find the volume of the region in three space bounded by the plane z=4 and the paraboloid z = x^2 + y^2.
Lecture 63 Play Video	Volume of a Snow Cone in Cylindrical and Spherical Coordinates Multivariable Calculus: Find the volume of the region above the xy-plane bounded between the sphere x^2 + y^2 + z^2 = 16 and the cone z^2 = x^2 + y^2. We solve in both cylindrical and spherical coordinates. We show how to derive the volume element by finding the volume of a spherical cube.
Lecture 64 Play Video	Example of Vector Field Multivariable Calculus: Sketch the vector field F(x, y) = ( y/sqrt(x^2+y^2), -x/sqrt(x^2+y^2) ) at the points (0,1), (4,3), (0,1), and (-3,4). We note some interesting properties of this vector field after switching to polar coordinates.
Lecture 65 Play Video	Example of Arc Length Along a Parametrized Curve Multivariable Calculus: Find the arc length of the parametrized curve r(t) = (2sqrt(2)/3 t^{3/2}, 1/2 t^2, t+3) from t=0 to t=2.
Lecture 66 Play Video	Sketching a Parametrized Curve Multivariable Calculus: Sketch the curve for the vector-valued function r(t) = (cos(t), sin(t), -cos(t)-sin(t) +1). We describe the trace of the curve as the intersection of a cylinder and a plane.
Lecture 67 Play Video	Line Integral of xy^3 over Unit Circle in Q1 Multivariable Calculus: Compute the integral of xy^3 over the unit circle in the first quadrant (counterclockwise).