Multivariable calculus
Learn multivariable calculus—derivatives and integrals of multivariable functions, application problems, and more.
5 units177 videos25 hours
Learn it live with a class on Puzzicle · Teachers: use it with your class
Unit 1: Thinking about multivariable functions
Introduction to multivariable calculus
Visualizing scalar-valued functions
Visualizing vector-valued functions
Unit 2: Derivatives of multivariable functions
Partial derivatives
Gradient and directional derivatives
Gradient ✓ Check-ins ✎ Fixed a slip5:31
Gradient and graphs ✓ Check-ins6:11
Gradient and contour maps ✓ Check-ins ✎ Fixed a slip6:17
Directional derivative ✓ Check-ins7:14
Directional derivative, formal definition ✓ Check-ins6:39
Directional derivatives and slope ✓ Check-ins ✎ Fixed a slip8:50
Why the gradient is the direction of steepest ascent ✓ Check-ins ✎ Fixed a slip10:32
Differentiating parametric curves
Multivariable chain rule
Curvature
Curvature intuition ✓ Check-ins4:12
Curvature formula, part 1 ✓ Check-ins6:06
Curvature formula, part 2 ✓ Check-ins7:06
Curvature formula, part 3 ✓ Check-ins8:37
Curvature formula, part 4 ✓ Check-ins8:34
Curvature formula, part 5 ✓ Check-ins7:46
Curvature of a helix, part 1 ✓ Check-ins6:28
Curvature of a helix, part 2 ✓ Check-ins7:19
Curvature of a cycloid ✓ Check-ins4:43
Partial derivatives of vector-valued functions
Computing the partial derivative of a vector-valued function ✓ Check-ins2:18
Partial derivative of a parametric surface, part 1 ✓ Check-ins9:29
Partial derivative of a parametric surface, part 2 ✓ Check-ins7:03
Partial derivatives of vector fields ✓ Check-ins8:34
Partial derivatives of vector fields, component by component ✓ Check-ins ✎ Fixed a slip7:31
Divergence
Curl
2d curl intuition ✓ Check-ins ✎ Fixed a slip3:54
2d curl formula ✓ Check-ins5:43
2d curl example ✓ Check-ins5:39
2d curl nuance ✓ Check-ins5:00
Describing rotation in 3d with a vector ✓ Check-ins6:06
3d curl intuition, part 1 ✓ Check-ins ✎ Fixed a slip5:50
3d curl intuition, part 2 ✓ Check-ins8:17
3d curl formula, part 1 ✓ Check-ins7:41
3d curl formula, part 2 ✓ Check-ins ✎ Fixed a slip6:12
3d curl computation example ✓ Check-ins5:35
Laplacian
Unit 3: Applications of multivariable derivatives
Tangent planes and local linearization
Quadratic approximations
What do quadratic approximations look like ✓ Check-ins4:42
Quadratic approximation formula, part 1 ✓ Check-ins7:09
Quadratic approximation formula, part 2 ✓ Check-ins9:50
Quadratic approximation example ✓ Check-ins ✎ Fixed a slip12:09
The Hessian matrix ✓ Check-ins6:10
Expressing a quadratic form with a matrix ✓ Check-ins8:20
Vector form of multivariable quadratic approximation ✓ Check-ins ✎ Fixed a slip8:35
Optimizing multivariable functions
Multivariable maxima and minima ✓ Check-ins8:04
Saddle points ✓ Check-ins ✎ Fixed a slip5:50
Warm up to the second partial derivative test ✓ Check-ins11:37
Second partial derivative test ✓ Check-ins ✎ Fixed a slip11:52
Second partial derivative test intuition ✓ Check-ins10:43
Second partial derivative test example, part 1 ✓ Check-ins6:11
Second partial derivative test example, part 2 ✓ Check-ins ✎ Fixed a slip7:43
Lagrange multipliers and constrained optimization
Constrained optimization introduction ✓ Check-ins ✎ Fixed a slip6:29
Lagrange multipliers, using tangency to solve constrained optimization ✓ Check-ins8:43
Finishing the intro lagrange multiplier example ✓ Check-ins6:39
Lagrange multiplier example, part 1 ✓ Check-ins7:50
Lagrange multiplier example, part 2 ✓ Check-ins6:51
The Lagrangian ✓ Check-ins ✎ Fixed a slip12:28
Meaning of the Lagrange multiplier ✓ Check-ins10:08
Proof for the meaning of Lagrange multipliers ✓ Check-ins17:19
Unit 4: Integrating multivariable functions
Line integrals for scalar functions
Line integrals in vector fields
Line integrals and vector fields ✓ Check-ins ✎ Fixed a slip16:46
Using a line integral to find work ✓ Check-ins11:32
Parametrization of a reverse path ✓ Check-ins7:26
Scalar field line integral independent of path direction ✓ Check-ins16:22
Vector field line integrals dependent on path direction ✓ Check-ins ✎ Fixed a slip15:51
Path independence for line integrals ✓ Check-ins17:14
Closed curve line integrals of conservative vector fields ✓ Check-ins8:25
Example of closed line integral of conservative field ✓ Check-ins11:02
Second example of line integral of conservative vector field ✓ Check-ins ✎ Fixed a slip10:19
Double integrals
Double integrals (articles)
Triple integrals
Surface integral preliminaries
Surface integrals
Introduction to the surface integral ✓ Check-ins ✎ Fixed a slip22:14
Example of calculating a surface integral part 1 ✓ Check-ins10:45
Example of calculating a surface integral part 2 ✓ Check-ins10:29
Example of calculating a surface integral part 3 ✓ Check-ins ✎ Fixed a slip9:54
Surface integral example, part 1 ✓ Check-ins11:05
Surface integral example part 2 ✓ Check-ins12:51
Surface integral example part 3: The home stretch ✓ Check-ins ✎ Fixed a slip12:39
Surface integral ex2 part 1 ✓ Check-ins5:10
Surface integral ex2 part 2 ✓ Check-ins9:51
Surface integral ex3 part 1 ✓ Check-ins9:00
Surface integral ex3 part 2 ✓ Check-ins9:10
Surface integral ex3 part 3 ✓ Check-ins12:41
Surface integral ex3 part 4 ✓ Check-ins4:02
Unit 5: Green's, Stokes', and the divergence theorems
Green's theorem
2D divergence theorem
Stokes' theorem
Stokes' theorem intuition ✓ Check-ins12:12
Green's and Stokes' theorem relationship ✓ Check-ins6:55
Orienting boundary with surface ✓ Check-ins3:29
Orientation and stokes ✓ Check-ins4:26
Conditions for stokes theorem ✓ Check-ins4:44
Stokes example part 1 ✓ Check-ins3:10
Stokes example part 2 ✓ Check-ins ✎ Fixed a slip4:02
Stokes example part 3 ✓ Check-ins8:05
Stokes example part 4 ✓ Check-ins6:54
Evaluating line integral directly - part 1 ✓ Check-ins7:44
Evaluating line integral directly - part 2 ✓ Check-ins ✎ Fixed a slip6:23
3D divergence theorem
Proof of Stokes' theorem
Stokes' theorem proof part 1 ✓ Check-ins5:14
Stokes' theorem proof part 2 ✓ Check-ins7:39
Stokes' theorem proof part 3 ✓ Check-ins4:11
Stokes' theorem proof part 4 ✓ Check-ins7:19
Stokes' theorem proof part 5 ✓ Check-ins5:28
Stokes' theorem proof part 6 ✓ Check-ins5:27
Stokes' theorem proof part 7 ✓ Check-ins ✎ Fixed a slip11:49
Types of regions in three dimensions
Divergence theorem proof
Learn this with a class
With a class on Puzzicle this course runs as a timetable of live study parties: your batch watches together and answers the check-ins, sometimes with a teacher leading.
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