Calc Exam 2

condition for volume integral

f(x,y)>=0

volume under solid z=f(x,y) over the region R

Fubini’s Theorem

Volume Integral over solid E

Jacobian of transformation for (u,v)

change of variables polar coordinates

Jacobian= r

Jacobian in 3d for (u,v,w)

change of variables cylindrical coordinates

jacobian= r

spherical coordinates change of variables

spherical coordinates jacobian

Line integral of scalar functions conditions

let c be a c1 curve and f be a scalar function defined over C, f:C→ R

line integrals of scalar functions

line integrals of vector fields conditions

F is a C1 vector field and C is a C1 curve

line integral of vector fields

If an object is moving on a curve C under a force F the work after time [a,b]

Fundamental Theorem for Line Integrals

f is a conservative vector field if (r3)

we can find an F such that F=gradient of f

curl of F

If the curl of f is zero the field is conservative

component test( showing F is conservative on R2)

if F=(P,Q) is a C1 vector field defined on an open simply connected region D that satisfies the equation then F is conservative

Greens Theorem Conditions

Let C be a positively oriented, peicewise smooth, simple, closed curve in the plane

Greens Theorem

If a curve C is traversed clockwise

If C is traversed n times

the line integral is multiplied by n

Greens Theorem and Area

If Qdx-Pdy= 1 then it can be used to find area

normal vector on a surface at each point (u,v)

Surface area of a parametrized surface r(u,v)

Surface Integrals of Scalar functions

positive orientation

outward/upward

closed surfaces are given the _____ orientation

positive/outward

Surface Integral for vector field

where the sign is fixed by the chosen orientation

Stokes Theorem Conditions

Let S be an oriented, piecewise smooth surface bounded by a simple closed curve C with the induced positive orientation. F is a C1 vector field on s

Stokes Theorem

Gauss’ Divergence Theorem Conditions

E a simple solid region in R3 and S be its closed boundary surface oriented outward. F is a C1 vector field on E

Gauss’ Divergence Theorem