# Calculus III: Vector Calculus & Differential Equations

### Course Description

In this course, Krista King from the integralCALC Academy covers a range of topics in Multivariable Calculus, including Vectors, Partial Derivatives, Multiple Integrals, and Differential Equations.

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### Video Lectures & Study Materials

Visit the official course website for more study materials: https://www.youtube.com/playlist?list=PLJ8OrXpbC-BMdeuQfJDVRJ5DPMduSzVow

# Lecture Play Lecture
I. Partial Derivatives
1 Partial Derivatives (007:33) Play Video
2 Second Order Partial Derivatives (007:58) Play Video
3 Equation of the Tangent Plane in Two Variables (005:53) Play Video
4 Normal Line to the Surface (0011:24) Play Video
5 Linear Approximation in Two Variables (006:05) Play Video
6 Linearization of a Multivariable Function (007:03) Play Video
7 Differential of the Multivariable Function (004:43) Play Video
8 Chain Rule for Partial Derivatives of Multivariable Functions (0015:00) Play Video
9 Chain Rule and Tree Diagrams of Multivariable Functions (008:47) Play Video
10 Implicit Differentiation for Partial Derivatives of Multivariable Functions (008:33) Play Video
11 Directional Derivatives (007:41) Play Video
12 Gradient Vectors (005:00) Play Video
13 Gradient Vectors and the Tangent Plane (006:37) Play Video
14 Gradient Vectors and Maximum Rate of Change (006:06) Play Video
15 Second Derivative Test: Two Variables (0010:09) Play Video
16 Local Extrema and Saddle Points of a Multivariable Function (0011:26) Play Video
17 Global Extrema in Two Variables (008:43) Play Video
18 Extreme Value Theorem and Extrema in the Set D (0018:49) Play Video
19 Max Product of Three Real Numbers (0013:18) Play Video
20 Max Volume of a Rectangular Box Inscribed in a Sphere (0015:30) Play Video
21 Points on the Cone Closest to a Point (008:51) Play Video
II. Lagrange Multipliers
22 Lagrange Multipliers (Part I) (009:29) Play Video
23 Lagrange Multipliers (Part II) (007:30) Play Video
24 Lagrange Multipliers in Three Dimensions with Two Constraints (0014:57) Play Video
III. Double Integrals
25 Midpoint Rule to Approximate Volume of a Double Integral (009:33) Play Video
26 Riemann Sums to Approximate Volume of a Double Integral (008:50) Play Video
27 Average Value of a Double Integral (006:59) Play Video
28 Iterated Integrals (009:05) Play Video
29 Double Integrals (007:33) Play Video
30 Double Integrals of Type I and Type II Regions (0012:19) Play Video
31 Double Integrals to Find the Volume of the Solid (008:49) Play Video
32 Double Integrals to Find Surface Area (0012:15) Play Video
33 Converting Iterated Integrals to Polar Coordinates (0010:51) Play Video
34 Converting Double Integrals to Polar Coordinates (0012:52) Play Video
35 Sketching the Region Given by a Double Polar Integral (005:54) Play Video
36 Double Polar Integral to Find Area (0012:19) Play Video
37 Double Polar Integral to Find the Volume of the Solid (0012:34) Play Video
38 Double Integrals to Find Mass and Center of Mass of the Lamina (0012:12) Play Video
IV. Triple Integrals
39 Midpoint Rule for Triple Integrals (0011:58) Play Video
40 Average Value of the Triple Integral (006:39) Play Video
41 Triple Iterated Integrals (0010:37) Play Video
42 Triple Integrals (0013:43) Play Video
43 Triple Integrals to Find Volume of the Solid (0014:06) Play Video
44 Expressing a Triple Iterated Integral Six Ways (0018:18) Play Video
45 Mass and Center of Mass with Triple Integrals (0011:24) Play Video
46 Moments of Inertia with Triple Integrals (008:12) Play Video
47 Cylindrical Coordinates (004:15) Play Video
48 Converting Triple Integrals to Cylindrical Coordinates (0013:55) Play Video
49 Volume in Cylindrical Coordinates (0012:23) Play Video
50 Spherical Coordinates (003:57) Play Video
51 Triple Integral in Spherical Coordinates to Find Volume (008:36) Play Video
52 Jacobian of the Transformation (2x2) (006:18) Play Video
53 Jacobian of the Transformation (3x3) (009:42) Play Video
54 Plotting Points in Three Dimensions (0010:56) Play Video
V. Vector Calculus & Conic Sections
55 Distance Formula for Three Variables (0010:25) Play Video
56 Equation of a Sphere, Plus Center and Radius (0010:06) Play Video
57 Describing a Region in 3D Space (005:14) Play Video
58 Using Inequalities to Describe a Region in 3D Space (005:56) Play Video
59 Finding a Vector From Two Points (002:46) Play Video
60 Vector Addition and Combinations of Vectors (007:52) Play Video
61 Sum of Two Vectors (002:38) Play Video
62 Copying Vectors to Find Combinations of Vectors (009:45) Play Video
63 Unit Vector in the Direction of the Given Vector (005:32) Play Video
64 Angle Between a Vector and the x-axis (008:12) Play Video
65 Magnitude and Angle of the Resultant Force (0013:02) Play Video
66 Dot Product of Two Vectors (003:30) Play Video
67 Angle Between Two Vectors (005:30) Play Video
68 Orthogonal, Parallel or Neither (Vectors) (007:00) Play Video
69 Acute Angle Between the Lines (Vectors) (009:01) Play Video
70 Acute Angles Between the Curves (Vectors) (0017:07) Play Video
71 Direction Cosines and Direction Angles (Vectors) (008:40) Play Video
72 Scalar Equation of a Line (002:30) Play Video
73 Scalar Equation of a Plane (003:07) Play Video
74 Scalar and Vector Projections (007:42) Play Video
75 Cross Product (007:45) Play Video
76 Vector Orthogonal to the Plane (009:12) Play Video
77 Volume of the Parallelepiped Determined by Vectors (007:05) Play Video
78 Volume of the Parallelepiped with Adjacent Edges (008:21) Play Video
79 Scalar Triple Product to Verify the Vectors are Coplanar (008:56) Play Video
80 Vector and Parametric Equations of the Line (006:51) Play Video
81 Parametric and Symmetric Equations of the Line (008:47) Play Video
82 Symmetric Equations of a Line (002:48) Play Video
83 Parallel, Intersecting, Skew and Perpendicular Lines (0010:41) Play Video
84 Equation of the Plane Using Vectors (008:09) Play Video
85 Point of Intersection of a Line and a Plane (004:22) Play Video
86 Parallel, Perpendicular, and Angle Between Planes (009:30) Play Video
87 Parametric Equations for the Line of Intersection of Two Planes (0012:38) Play Video
88 Symmetric Equations for the Line of Intersection of Two Planes (0010:52) Play Video
89 Distance Between a Point and a Line (Vectors) (008:55) Play Video
90 Distance Between a Point and a Plane (Vectors) (007:20) Play Video
91 Distance Between Parallel Planes (Vectors) (008:35) Play Video
92 Sketching the Quadric Surface (008:08) Play Video
93 Reducing a Quadric Surface Equation to Standard Form (0018:13) Play Video
94 Domain of the Vector Function (005:19) Play Video
95 Limit of the Vector Function (006:01) Play Video
96 Sketching the Vector Equation (0011:04) Play Video
97 Projections of the Curve Onto the Coordinate Axes (0016:37) Play Video
98 Vector and Parametric Equations of the Line Segment (005:09) Play Video
99 Vector Function for the Curve of Intersection of Two Surfaces (005:45) Play Video
100 Derivative of the Vector Function (008:02) Play Video
101 Unit Tangent Vector (006:27) Play Video
102 Parametric Equations of the Tangent Line (Vectors) (008:26) Play Video
103 Integral of the Vector Function (009:09) Play Video
104 Green's Theorem: One Region (008:28) Play Video
105 Green's Theorem: Two Regions (0016:31) Play Video
VI. First Order Differential Equations
106 Linear Differential Equations (009:13) Play Video
107 Circuits and Linear Differential Equations (007:41) Play Video
108 Linear Differential Equation Initial Value Problem (0010:11) Play Video
109 Differential Equations (006:57) Play Video
110 Change of Variable to Solve a Differential Equations (005:09) Play Video
111 Separable Differential Equations Initial Value Problem (006:19) Play Video
112 Mixing Problems with Separable Differential Equations (0011:17) Play Video
113 Euler's Method (Part I) (009:29) Play Video
114 Euler's Method (Part II) (009:56) Play Video
115 Euler's Method (Part III) (005:13) Play Video
116 Sketching Direction Fields (008:40) Play Video
117 Population Growth (006:07) Play Video
118 Logistic Growth Model of a Population (006:29) Play Video
119 Predator-Prey Systems (0013:41) Play Video
VII. Second Order Differential Equations
120 Second-Order Differential Equations (002:58) Play Video
121 Equal Real Roots of Second-Order Homogeneous Differential Equations (004:41) Play Video
122 Complex Conjugate Roots of Second-Order Homogeneous Differential Equations (008:24) Play Video
123 Second-Order Differential Equations: Initial Value Problems (Example 1) (0010:31) Play Video
124 Boundary Value Problem, Second-Order Homogeneous Differential Equation, and Distinct Real Roots (009:26) Play Video
125 Boundary Value Problem, Second-Order Homogeneous Differential Equation, and Complex Conjugate Roots (007:52) Play Video
126 Second-Order Differential Equations: Working Backwards (003:45) Play Video
127 Second-Order Non-Homogeneous Differential (0021:19) Play Video
128 Variation of Parameters for Differential Equations (0013:15) Play Video
129 Second-Order Non-Homogeneous Differential Equations: Initial Value Problem (0024:20) Play Video
VIII. Laplace Transforms
130 Laplace Transforms Using the Definition (0013:47) Play Video
131 Laplace Transforms Using a Table (004:31) Play Video
132 Initial Value Problems with Laplace Transforms (0020:47) Play Video
133 Laplace Transforms and Integration by Parts with Three Functions (0026:02) Play Video
134 Inverse Laplace Transform (0010:45) Play Video
135 Convolution Integral for Initial Value Problems (0017:44) Play Video
136 Exact Differential Equations (0016:43) Play Video
IX. Lagrange Multipliers
137 Lagrange Multipliers and Three Dimensions, One Constraint (008:36) Play Video
138 Limit of the Multivariable Function (006:47) Play Video
139 Minimum Distance Between the Point and the Plane (004:43) Play Video
140 Precise Definition of the Limit for Multivariable Functions (0034:23) Play Video
141 Critical Points of Multivariable Functions (005:28) Play Video
142 Discontinuities of a Multivariable Function (004:11) Play Video
143 Domain of a Multivariable Function (005:40) Play Video
144 Arc Length of a Vector Function (009:42) Play Video
145 Area of the Surface (0010:54) Play Video
146 Tangential and Normal Components of the Acceleration Vector (0013:58) Play Video
X. Line Integrals
147 Curl and Divergence (0012:24) Play Video
148 Curvature of the Vector Function (0011:49) Play Video
149 Independence of Path (0015:52) Play Video
150 Line Integral of a Curve (0016:29) Play Video
151 Line Integral of a Vector Function (0010:42) Play Video
152 Maximum Curvature of the Function (0013:11) Play Video
153 Normal and Osculating Planes (0022:56) Play Video
154 Parametric Representation of the Surface (008:32) Play Video
155 Points on the Surface (007:24) Play Video
156 Potential Function of a Conservative Vector Field (0013:01) Play Video
157 Potential Function of the Conservative Vector Field to Evaluate a Line Integral (0013:36) Play Video
158 Potential Function of the Conservative Vector Field, Three Dimensions (0017:37) Play Video
159 Re-parametrizing the Curve in Terms of Arc Length (008:00) Play Video