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- Math 251........The end

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Find the gradient vector field of f(scalar)

take the individual partials with respect to the funciton

Line integrals for scalar

1. parameterize the curve.

2. fid Ds (ds is the magnitude of the derivative of the parameterization of the curve)

3. now multiply ds by the function in terms of the parameterization.

4. solve that bitch

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line integrals for vectors

same steps as before but you dot Ds instead of taking the magnitude( you cant integrate vectors)

how to test for conservativeness

if dp/dy= dq/dx then the function is conservative.

find potential function

1. pick a part of the vector ( for example gx)

2. integrate with respect to X with the addition of the unknown Y

3. take the derivative with respect to Y

4. set equal to gy.

5.integrate with respect to Y

(note this only works if the field is conservative)

fundamental theorem of line integrals.

once the potential is found plug in the points. or in other words end - start

All line integrals

positive orientation is counter clockwise.

greens theorem

double integral of qx-py da

Only works if function is simple and closed and continuous partial derivatives

Steps

1.find qx and py

2. plug it in and put the bounds.

Curl

you should just know this shit and if curl = 0 it is conservative.

divergence

px + qy + rz

surface integrals (scalar)

if its not smooth use the double integral of sqrt( 1+dx2+dy2) times what ever function is given.

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Surface integrals for vector

1. parameterize the curve.

2.take partials of the parameterization.

3. cross them to get the normal.

4.dot that with the vector in terms of the parameterization.

5. integrate.

Stokes' Theorem

∫_{c} F • dr = ∫∫_{s} curl(F) • dS

its the flux of the curl

curve: r(t)

vector field F

surface S with boundary curve C

Divergence Theorem

∫_{S}∫F•NdS = ∫∫_{E}∫divFdV

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