Khan Academy on a Stick
Line integrals and Green's theorem
Line integral of scalar and vectorvalued functions. Green's theorem and 2D divergence theorem.

Introduction to the Line Integral
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Introduction to the Line Integral

Line Integral Example 1
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Concrete example using a line integral

Line Integral Example 2 (part 1)
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Line integral over a closed path (part 1)

Line Integral Example 2 (part 2)
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Part 2 of an example of taking a line integral over a closed path
Line integrals for scalar functions
With traditional integrals, our "path" was straight and linear (most of the time, we traversed the xaxis). Now we can explore taking integrals over any line or curve (called line integrals).

Position Vector Valued Functions
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Using a position vector valued function to describe a curve or path

Derivative of a position vector valued function
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Visualizing the derivative of a position vector valued function

Differential of a vector valued function
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Understanding the differential of a vector valued function

Vector valued function derivative example
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Concrete example of the derivative of a vector valued function to better understand what it means
Position vector functions and derivatives
In this tutorial, we will explore position vector functions and think about what it means to take a derivative of one. Very valuable for thinking about what it means to take a line integral along a path in a vector field (next tutorial).

Line Integrals and Vector Fields
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Using line integrals to find the work done on a particle moving through a vector field

Using a line integral to find the work done by a vector field example
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Using a line integral to find the work done by a vector field example

Parametrization of a Reverse Path
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Understanding how to parametrize a reverse path for the same curve.

Scalar Field Line Integral Independent of Path Direction
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Showing that the line integral of a scalar field is independent of path direction

Vector Field Line Integrals Dependent on Path Direction
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Showing that, unlike line integrals of scalar fields, line integrals over vector fields are path direction dependent

Path Independence for Line Integrals
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Showing that if a vector field is the gradient of a scalar field, then its line integral is path independent

Closed Curve Line Integrals of Conservative Vector Fields
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Showing that the line integral along closed curves of conservative vector fields is zero

Example of Closed Line Integral of Conservative Field
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Example of taking a closed line integral of a conservative field

Second Example of Line Integral of Conservative Vector Field
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Using path independence of a conservative vector field to solve a line integral
Line integrals in vector fields
You've done some work with line integral with scalar functions and you know something about parameterizing positionvector valued functions. In that case, welcome! You are now ready to explore a core tool math and physics: the line integral for vector fields. Need to know the work done as a mass is moved through a gravitational field. No sweat with line integrals.

Green's Theorem Proof Part 1
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Part 1 of the proof of Green's Theorem

Green's Theorem Proof (part 2)
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Part 2 of the proof of Green's Theorem

Green's Theorem Example 1
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Using Green's Theorem to solve a line integral of a vector field

Green's Theorem Example 2
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Another example applying Green's Theorem
Green's theorem
It is sometimes easier to take a double integral (a particular double integral as we'll see) over a region and sometimes easier to take a line integral around the boundary. Green's theorem draws the connection between the two so we can go back and forth. This tutorial proves Green's theorem and then gives a few examples of using it. If you can take line integrals through vector fields, you're ready for Mr. Green.

Constructing a unit normal vector to a curve
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Figuring out a unit normal vector at any point along a curve defined by a position vector function

2 D Divergence Theorem
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Using Green's Theorem to establish a two dimensional version of the Divergence Theorem

Conceptual clarification for 2D Divergence Theorem
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Understanding the line integral as flux through a boundary
2D Divergence theorem
Using Green's theorem (which you should already be familiar with) to establish that the "flux" through the boundary of a region is equal to the double integral of the divergence over the region. We'll also talk about why this makes conceptual sense.