Calculus: Vector Fields, Line integrals, Green's Theorem

It’s a collection of vectors each assigned to a point in a plane or space.

n inputs produces n component outputs of the vector at the point.

1. Go to the point to be plugged in to the function.

2. Evaluate the function with the points plugged in to derive the vector components.

3. Draw the vector at that point.

Vectors point away from or towards the origin.

In this case it’s vertical, but it could also be horizontal by setting the y-component to 0 and letting the x-component be equal to the y-point.

The potential function is a single function (phi).

The gradient field is a gradient- a vector -derived from taking the partial derivative with respect to each variable.

The gradient points perpendicular and in the direction of steepest increase.

Instead of integrating over an interval along an axis you are integrating over a curve or path ( C ).

Think of it like a curtain. The curve outlines the shape on the 2-D plane, the function gives it the sheet above the curve.

– parametrize the curve C

– Define x, y, and z in terms of t: x(t), y(t),z(t)

– Take the derivative of each: x'(t), y'(t), z'(t)

– Determine the range of t

– for what you define x, y, and z in terms of t: substitute each into the function f(x,y,z) in the integrand.

This is essentially considering a scalar line integral path while a vector field interacts with it in the background.

dr in this case is a unit tangent vector as is the vector field function. The dot product of the two relates their magnitudes with respect to the angle between them all along C.

1. Integrate each component of the gradient field with respect to x, y and z respectively.

2. Be sure to add a constant function in terms of the other variables for each integral. C(y,z), C(x,z) and C(x,y) respectively.

3. Look at each integral and determine which functions could’ve belonged in the original potential function.

4. Remember to account for the constant functions that could’ve been eliminated completely when taking the partial with respect to each variable.

ds = | r'(t) | (which is the base)

Height = f(x,y,z)

base x height over the interval:
a ≤ t ≤ b

Note by “path parametrization”, this means orientation or direction. Which matters in VECTOR Line integrals.

A shortcut for computing a vector line integral.

(potential function evaluated at c(t) of the end point) – (potential function evaluated at ct) of the start point )

Make sure it’s a CONSERVATIVE Vector Field

Basically if the vector field is a gradient of some function, it is conservative.

P and Q are the respective component functions in the Vector Field function F.

Note the CLOSED curve enclosing the region is oriented COUNTERCLOCKWISE