math final

greens theorem

how to convert a integral over a curvy path into a standard integral

ds=|r’(t)|dt

div grad(f) f is scalar

grad= input=scalar output=vector

div= input=vector output=scalar

scalar

curl (f) f is scalar

curl = input=vector output=vector

not meaningful

grad(F*F) F is vector

dot product of 2 vector is scalar

grad= input=scalar output=vector

vector

div(fF)

dot product is vector

div= input=vector output=scalar

scalar

curl grad(f) f is scalar

grad= input=scalar output=vector

curl=input=vector output=vector

vector

grad div (F) F is vector

grad= input=scalar output=vector

div= input=vector output=scalar

vector

understanding integration order dydzdx

inner bounds dy can depend on x and z

middle bounds dz can only depend on x

outer bounds dx must be constants

div(curl F)=0

if G=curlF then div G=0

how to convert to surface integral z=g(x,y)

ds=(-g_x,-g_y,1)dxdy