MAT 21D Final

How to find potential function of a vector field

Integrate P with respect to x to find f(x, y, z) Integrate Q with respect to y to find g(x, y, z) Integrate R with respect to z to find h(x, y, z) Combine f, g, and h to form the potential function F(x, y, z) = f(x, y, z) + g(x, y, z) + h(x, y, z) + C confirm that ∇F = F

path independent and conservative

Let F be a vector field defined on an open region D in space. Suppose that for any two points A and B in D, the path C from A to B in D is the same over all paths from A to B.

Moments of inertia

M = ∫δ Mx = ∫yδ My = ∫xδ x (center of mass) = My / M y (center of mass) = Mx/M

How to show that vector field is conservative

F(x,y,z) = Mi + Nj + Pk calculate curl (∇ x F) - if curl(F) = 0, vector field is conservative ^^ can also use this method to calculate the line integral. plug coordinates into initial (F) function and substract

Curl of a vector field

(Ry−Qz)i + (Pz−Rx)j + (Qx−Py)k

cross product

(u2v3 - u3v2)i + (u1v3 - u3v1)j + (u1v2 - u2v1)k

Find arc length parametrization

1. find the arc length (s =xt) 2. solve for t 3. plug back into position equation to get r(s)

average value of a function

1/area ∫∫f(x,y)

Surface integral of vector field

G(x,y,f(x,y) sqrt(fx^2 + fy^2 + 1)

Find the work done by F in moving an object along a smooth curve C

If F is a conservative field: W = f(B) - f(A) curlF = 0 W = ∫F(r(t)) * r'(t)

spherical coordinates

are used to locate points on a spherical surface by specifying two angles and one distance x=ρsinφcosθ y=ρsinφsinθ z=ρcosφ ρ^2=x^2+y^2+z^2

unit binomial vector

cross product of tangent and normal

tangential component of acceleration

d/dt | v(t) |

F is conservative if:

dP/dy = dQ/dx

line integrals of line segment C

r(t) = (1-t)<x,y,z> + (t)<x,y,z>

normal component of acceleration

sqrt[ |a|^2 - (aT)^2 ]

When are two vectors parallel?

the cross product is 0

scalar line integrals

the integrand is a function of more than one variable - domain of integration is a curve in a plane/space

vectors are perpendicular if:

u1v1 + u2v2 + u3v3 = 0 (dot product)

cylindrical coordinates

x=rcosθ y=rsinθ z=z r^2=x^2+y^2

Surface area of a bounded region

|∇F| / |∇F * p| p is normal to region

potential function f(x,y)

∇f = Pi + Qj = F f(x,y) = ∫P(x,y) dx = ___ + h(y) find df/fy = Q to find h(y)

Flux

∫ Mdy - Ndx

integration by parts

∫ u dv = uv - ∫ v du

Flow of a velocity field

∫ velocity field * dr

Vector Line Integral

∫Px' + Qy' + Rz'

Stokes' Theorem

∫∫(∇xF)(x,y,f(x,y))