# Calculus: Vector Fields, Line integrals, Green's Theorem It’s a collection of vectors each assigned to a point in a plane or space.
Vector Field (Graph definition) n inputs produces n component outputs of the vector at the point.
Vector Field (Function definition)
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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.

How is a vector field plotted? Rotation Vector Field 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.
Shear Flow Vector Field 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.
Graphically what is the relationship between the gradient field and potential function? 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.

SCALAR line integral (visual)? – 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.

What are the steps for evaluating a SCALAR Line Integral? 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.

VECTOR Line Integral (Visual)?
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.

How is a potential function found for a gradient field? ds = | r'(t) | (which is the base)

Height = f(x,y,z)

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

Formula for evaluating a SCALAR Line Integral? How is it derived? Note by “path parametrization”, this means orientation or direction. Which matters in VECTOR Line integrals.
What are the main differences between Scalar and Vector Line Integrals? Formula for a VECTOR Line Integral? What is ds and dr equal to in the scalar and vector line integral respectively? 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

Fundamental Theorem of Line Integrals? Simple and Closed Curves Connected and Simply Connected Curves Basically if the vector field is a gradient of some function, it is conservative.
Conservative Vector Field Piecewise Smooth Curve Integral notation for a closed path is given by? What is the shortcut test to tell if a Vector field 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

Greene’s Theorem?

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