u3 processes

Surface (area) Integral

1. parametrize

2. get partials of vector for both vars

3. cross the partials

4. get magnitude

5. integrate over bounds

Flux

1. parametrize

2. find n(t) = <y’(t),-x’(t)>

3. evaluate F(r(t))

4. dot F(r(t)) with n(t)

5. integrate over bounds

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Green’s Theorem:

If positively oriented, simple, closed curve C bounding region D:

<P,Q> = F
Double integral D of (∂P/∂x + ∂Q/∂y) dA

• If integrand becomes constant, just multiply by the area

conservative test

3d:
Construct 3×3 vector matrix with ijk,partials, and then components
Components should be <0,0,0>

2d:
y partial of first component should equal x partial of second component

When to find the potential function

• fundamental theorem

• computing work

• integral F * dr over C

• testing path independence

• evaluating line integrals quickly

• when the curve is complicated but the endpoints are easy

Divergence

Sum of partials

Curl

Vector result from the conservative test

Cross-product determinant

Work

If parameterized: integral of F(r(t)) * r’(t) dt from a to b
(line integral)

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If conservative: f(B)-f(A) using the potential function

Line integrals of vector fields

Integral F(r(t)) * r’(t) dt from a to b

Line integrals of scalar functions

Integral of f(r(t)) * ||r’(t)|| dt from a to b

Circulation

If parameterized: integral of F(r(t)) * r’(t) dt from a to b
(line integral)

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If conservative: f(B)-f(A) using the potential function

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Green’s Theorem:

If positively oriented, simple, closed curve C bounding region D:

<P,Q> = F

Circulation: Double integral D of (∂Q/∂x - ∂P/∂y) dA

Wire mass

Integral of ρ(r(t)) * ||r’(t)|| dt from a to b

Fundamental Theorem

If conservative, use potential function and do f(B)-f(A)

Decision Rule

If you see

integral C of F * dr →

• Is conservative? Use potential

• No? Parameterize

• Closed curve in 2D → Green’s

integral C of f ds → f(r(t))*||r’(t)||

Flux Problem:

• 2D closed curve: Green’s

• 3D surface: cross-product method