Line Integrals
Write the formula for the line integral of a vector field F on a curve C parametrized by r(t) from a<t<b.
∫C F dr
∫ab F(r(t)) · r'(t) dt
Write the formula for the surface integral of a function f(x,y,z) on a surface S parametrized by r(u,v) over the domain a<u<b and g(u)<v<h(u).
∫∫S f(x,y,z) dS
∫ab ∫g(u)h(u) f(r(u,v)) ||ru x rv|| dv du
Parametrize the circle y2+z2 = 16 and x=3 that is oriented counter-clockwise when looking down the positive x-axis towards the origin.
r(t) = <3, 4cos(t), 4sin(t) >
0<t<2π
Which theorem equates the circulation around a simple, closed, curve in R2 with the double integral of the z-component of curl over the region inside of C.
Green's Theorem for Circulation
Find the divergence of the vector field
F = < x2+3yz, cos(x2) + yz, z - exy >
2x+z+1
Write the formula for the line integral of a function f(x,y,z) on a curve C parametrized by r(t) from a<t<b.
∫C f(x,y,z) ds
∫ab f(r(t)) ||r'(t)|| dt
Write the formula for the total flux of a vector field F(x,y,z) on a surface S parametrized by r(u,v) over the domain a<u<b and g(u)<v<h(u).
∫∫S F · n dS
∫ab ∫g(u)h(u) F(r(u,v)) · (ru x rv) dv du
Parametrize the sphere of radius 3 centered at the origin.
r(u,v) = < 3cos(u)sin(v), 3sin(u)sin(v), 3cos(v) >
0<u<2π
0<v<π
Which theorem uses the chain rule for curves to provide a shortcut when computing line integrals on conservative vector fields?
Find the curl of the vector field
F = < 3x2, z3, 2yz >
< -x2, 0, 0 >
Calculate the circulation around r(t) = <cos(t), sin(t)> for 0<t<2π for the vector field F = <x,-y>.
∫02π <cos(t),-sin(t)> dot <-sin(t), cos(t)> dt
∫02π -2 cos(t) sin(t) dt
cos2(t) |02π = 0
Or by FTLI
Write the formula for the surface area of a surface S parametrized by r(u,v) over the domain a<u<b and g(u)<v<h(u).
∫ab ∫g(u)h(u) ||ru x rv|| dv du
Parametrize the cylinder with radius 2 centered around the z-axis between z=1 and z=5.
r(u,v) = <2cos(u), 2sin(u), v>
0<u<2π
1<v<5
Use Green's Theorem for Flux to find the flux out of the unit circle for the vector field
F = < 3x+y2 cos(y), 3x2 - y >
∫02π∫01 (3-1) r dr dθ =
∫02π∫01 2r dr dθ =
∫02π r2 |01 dθ =
∫02π 1 dθ = 2π
Is the following vector field conservative
F = < 3y2 + 2z , 6xy, 2x + ez >
Yes.
fxy = 6y =fyx
fzy = 0 =fyz
fxz = 2 =fzx
Let f(x,y) = 3xy - 2xy2 and let F be the gradient of f. Find the line integral of F over the curve C that is the union of C = C1 + C2 + C3
C1: The line from the origin to (0,5)
C2: The left half of the circle of radius 5 from (0,5) to (0,-5)
C3: The line from (0,-5) to (-1,-1)
Fundamental Theorem of Line Integrals
f(-1,-1) - f(0,0) = (3- -2) - (0-0) = 5
Daily Double:
Write Down all players on your team (First and Last Name) and a number between 1 and 3Everyone on your team gets that number of bonus points on the final exam.
Parametrize the paraboloid z= 3+x2+y2 between z=3 and z=12.
r(u,v) = < u, v, 3+u2+v2 >
0 < u2+v2 < 9
Use the divergence theorem to find the total flux flowing out of the closed cylinder x2 + y2 = 16 capped by the planes z=0 and z=5 for the vector field.
F = < 2x + 3yz - z2, 3xz - 3y + z3, x5y4 + 2z >
Div F = 2 - 3 + 2 = 1
The volume of a cylinder π r2 h
∫∫∫F 1 dV = π r2 h = π · 42 · 5 = 80π
Find the potential function for the vector field
F = < 3y2 + 2z , 6xy, 2x + ez >
f(x,y,z) = 3xy2 + 2xz + ez + C
Evaluate the line integral ∫C F dr where F is the vector field
F = <x2z, -y4, 10x3>
And C is the curve parametrized by
r(t)=< t, 2t, t2 >from 0<t<1.
F(r(t)) = < t4, -16t4, 10t3 >
r'(t) = < 1, 2, 2t >
∫01 < t4, -16t4, 10t3 > · < 1, 2, 2t > dt =
∫01 t4 - 32t4 + 20t4 dt =
∫01 -11t4 dt = -11/5
Calculate the surface integral of the function f(x,y,z) = 6 over the top half of the unit sphere centered at the origin.
∫ab ∫g(u)h(u) f(r(u,v)) ||ru x rv|| dv du =
∫02π ∫0π/2 6 (12 sin(φ)) dφ dθ =
∫02π ∫0π/2 6 sin(φ) dφ dθ=
∫02π -6cos(φ) |0π/2 dθ= ∫02π 6 dθ = 12 π
Parametrize the portion of the plane 4x + 2y + 6z =12 in the first octant.
r(u,v) = < u, 6-2u-3v, v >
0 < u < 3
0 < v < 2 - 2/3 u
(Alt. answers accepted)
Use Stokes' Theorem to find the circulation of the vector field
F = <6xy, z-x3, 3y-z>
around the curve y2+z2 = 16 and x=3 that is oriented counter-clockwise when looking down the positive x-axis towards the origin.
r(u,v) = <3, u, v > D: 0 < u2+v2 < 16
ru = <0,1,0> and rv= <0,0,1>
ru x rv = <1,0,0>
Curl F = < 2, 0, -3x2-6x >
∫∫S F · n dS = ∫∫D <2,0,-45>·<1,0,0> dA =
∫∫D 2 dA = 2(π·r2) =2(π · 42) = 32 π
Is there a vector field F such that the curl of F is
Why or why not
∇ · (∇ × F ) = 0 for all F. However
∇ · < 2yz, 3xz2, z - cos(xy) > = 0 + 0 + 1
Therefore, no vector field has curl equal to < 2yz, 3xz2, z - cos(xy) >