Khan Academy on a Stick : Surface integrals and Stokes' theorem

Khan Academy on a Stick

Calculus

Limits
Taking derivatives
Derivative applications
Indefinite and definite integrals
Solid of revolution
Sequences, series and function approximation
Double and triple integrals
Partial derivatives, gradient, divergence, curl
Line integrals and Green's theorem
Surface integrals and Stokes' theorem
Divergence theorem

Introduction to Parametrizing a Surface with Two Parameters
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Introduction to Parametrizing a Surface with Two Parameters
Determining a Position Vector-Valued Function for a Parametrization of Two Parameters
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Determining a Position Vector-Valued Function for a Parametrization of Two Parameters

Parameterizing a surface

You can parameterize a line with a position vector valued function and understand what a differential means in that context already. This tutorial will take things further by parametrizing surfaces (2 parameters baby!) and have us thinking about partial differentials.

Partial Derivatives of Vector-Valued Functions
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Partial Derivatives of Vector-Valued Functions
Introduction to the Surface Integral
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Introduction to the surface integral
Example of calculating a surface integral part 1
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Example of calculating a surface integral part 1
Example of calculating a surface integral part 2
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Example of calculating a surface integral part 2
Example of calculating a surface integral part 3
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Example of calculating a surface integral part 3
Surface Integral Example Part 1 - Parameterizing the Unit Sphere
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Visualizing a suitable parameterization
Surface Integral Example Part 2 - Calculating the Surface Differential
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Taking the cross product to calculate the surface differential in terms of the parameters
Surface Integral Example Part 3 - The Home Stretch
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Using a few trigonometric identities to finally calculate the value of the surface integral
Surface Integral Ex2 part 1 - Parameterizing the Surface
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Parametrizing a surface that can be explictly made a function of x and y.
Surface Integral Ex2 part 2 - Evaluating Integral
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Evaluating the surface integral
Surface Integral Ex3 part 1 - Parameterizing the Outside Surface
Breaking apart a larger surface into its components
Surface Integral Ex3 part 2 - Evaluating the Outside Surface
Evaluating the surface integral over the outside of the chopped cylinder
Surface Integral Ex3 part 3 - Top surface
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Parametrizing the top surface
Surface Integral Ex3 part 4 - Home Stretch
Evaluating the third surface integral and coming to the final answer

Surface integrals

Finding line integrals to be a bit boring? Well, this tutorial will add new dimension to your life by explore what surface integrals are and how we can calculate them.

Conceptual Understanding of Flux in Three Dimensions
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Conceptual understanding of flux across a two-dimensional surface
Constructing a unit normal vector to a surface
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Deriving a unit normal vector from the surface parametrization
Vector representation of a Surface Integral
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Different ways of representing a flux integral

Flux in 3-D and constructing unit normal vectors to surface

Flux can be view as the rate at which "stuff" passes through a surface. Imagine a next placed in a river and imagine the water that is flowing directly across the net in a unit of time--this is flux (and it would depend on the orientation of the net, the shape of the net, and the speed and direction of the current). It is an important idea throughout physics and is key for understanding Stokes' theorem and the divergence theorem.

Stokes' Theorem Intuition
Conceptual understanding of why the curl of a vector field along a surface would relate to the line integral around the surface's boundary
Green's and Stokes' Theorem Relationship
Seeing that Green's Theorem is just a special case of Stokes' Theorem
Orienting Boundary with Surface
Determining the proper orientation of the boundary given the orientation of the surface
Orientation and Stokes
Determining the proper orientation of a boundary given the orientation of the normal vector
Conditions for Stokes Theorem
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Understanding when you can use Stokes. Piecewise-smooth lines and surfaces
Stokes Example Part 1
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Starting to apply Stokes theorem to solve a line integral
Part 2 Parameterizing the Surface
Stokes Example Part 3 - Surface to Double Integral
Converting the surface integral to a double integral
Stokes Example Part 4 - Curl and Final Answer
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Finding the curl of the vector field and then evaluating the double integral in the parameter domain
Evaluating Line Integral Directly - Part 1
Showing that we didn't need to use Stokes' Theorem to evaluate this line integral
Evaluating Line Integral Directly - Part 2
Finishing up the line integral with a little trigonometric integration

Stokes' theorem intuition and application

Stokes' theorem relates the line integral around a surface to the curl on the surface. This tutorial explores the intuition behind Stokes' theorem, how it is an extension of Green's theorem to surfaces (as opposed to just regions) and gives some examples using it. We prove Stokes' theorem in another tutorial. Good to come to this tutorial having experienced the tutorial on "flux in 3D".

Stokes' Theorem Proof Part 1
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The beginning of a proof of Stokes' theorem for a special class of surfaces. Finding the curl of our vector field.
Stokes' Theorem Proof Part 2
Figuring out a parameterization of our surface and representing dS
Stokes' Theorem Proof Part 3
Writing our surface integral as a double integral over the domain of our parameters
Stokes' Theorem Proof Part 4
Starting to work on the line integral about the surface
Stokes' Theorem Proof Part 5
Working on the integrals...
Stokes' Theorem Proof Part 6
More manipulating the integrals...
Stokes' Theorem Proof Part 7
Using Green's Theorem to complete the proof

Proof of Stokes' theorem

You know what Stokes' theorem is and how to apply it, but are craving for some real proof that it is true. Well, you've found the right tutorial!