Test 4: 14.7 to 15.6

Question 1

In spherical coordinates, what is x equal to?

Answer 1

p sin phi cos theta

Question 2

In spherical coordinates, what is y equal to?

Answer 2

p sin phi sin theta

Question 3

In spherical coordinates, what is z equal to?

Answer 3

p cos phi

Question 4

In spherical coordinates what is p^2 equal to?

Answer 4

x^2 + y^2 + z^2

Question 5

In spherical coordinates what is tan theta equal to?

Answer 5

y/x

Question 6

In spherical coordinates, what is phi equal to (rect)?

Answer 6

arccos(z / sqrt(x^2+y^2+z^2))

Question 7

In spherical coordinates what is r^2 equal to?

Answer 7

p^2 sin^2 phi

Question 8

in spherical coordinates what is p equal to?

Answer 8

sqrt(r^2+z^2)

Question 9

In spherical coordinates what is phi equal to (cyl)?

Answer 9

arccos(z/sqrt(r^2+z^2))

Question 10

What is the integral template and order of integration for cylindrical coordinates?

Answer 10

intintint (Q) f(x,y,z) = intintint r dz dr dtheta

Question 11

What is the integral template and order of integration for spherical coordinates?

Answer 11

intintint (q) f(x,y,z) = intintint p^2 sin phi dp dphi dtheta

Question 12

How do you find the Jacobian?

Answer 12

It is the determinant of the partials of x,y (rows) with respect to (u,v) - px/pupy/pv - py/pupx/pv

Question 13

How do you find the change of variables for a Jacobian?

Answer 13

Set u = to one set of provided equations, and set v = to the other, solve for x and y in terms of u and v. If no equations provided, determine them by looking at the graph.

Question 14

How do you determine the u/v coordinates for a Jacobian?

Answer 14

Use the initial 4 equations and set u and v equal to their respective results.

Question 15

How do you find the integral for a Jacobian?

Answer 15

After determining the change of variables, plug x=u and y=v terms into the x and y function in the integral, and integrate over the region on the UV graph, multiplying by the Jacobian.

Question 16

How do you determine if a vector field is conservative?

Answer 16

Assign M, N, and P to each function to be integrated (x,y,z). Compare Mz with Px, Nz with Py, and My with Nx.

Question 17

How do you find the potential function for a vector field?

Answer 17

Integrate M dx, N dy, and P dz and combine the results, omitting duplicate expressions. Add +C.

Question 18

How do you find the curl of a vector field?

Answer 18

Find the del operator cross the vector field’s MNP. Row 1: i j k Row 2: p/x p/y p/z Row 3: M N P. Find determinant.

Question 19

How do you find the divergence of a vector field?

Answer 19

Find the del operator dot the vector field’s MNP: pM/x + pN/y + pP/z. Plug in point.

Question 20

How do you find a line integral of a non-vector field using ds format?

Answer 20

If C is given by r(t) = x(t)i + y(t)j, intC f(x,y) ds = int f(x(t), y(t), z(t)) * sqrt(x’^2 + y’^2 + z’^2) dt. Replace x, y and z with the appropriate parameterization using t.

Question 21

For parameterizations given by r(t) = x(t)i + y(t)j + z(t)k, what is ds?

Answer 21

||r’(t)|| dt = sqrt(x’(t)^2 + y’(t)^2 + z’(t)^2).

Question 22

How do you find a line integral of a vector field using dr format?

Answer 22

F dot r’(t) dt = F dot dr = int F(x(t), y(t), z(t)) dot r’(t) dt. Replace x, y and z with the appropriate parameterization using t.

Question 23

How do you write F dot dr in differential form?

Answer 23

intC M dx + N dy + P dz. Replace x, y and z with the appropriate parameterization using t.

Question 24

How do you use the fundamental theorem of line integrals?

Answer 24

Smooth curve, conservative, continuous. Integrate M dx, N dy and P dz. Plug end values for x,y,z into resulting function, subtract start values. No +C is necessary.

Question 25

What is Green’s Theorem?

Answer 25

COUNTERCLOCKWISE. intC M dz + N dy = intintR (pN/x - PM/y) dA

Question 26

What is the formula for the line integral for area?

Answer 26

A = 1/2 intC x dy - y dx

Question 27

What is the equation for a parametric surface?

Answer 27

r(u,v) = x(u,v)i + y(u,v)j + z(u,v)k

Question 28

How do you find the normal vector to a parametric surface?

Answer 28

By finding ru cross rv (ijk, px/u py/u pz/u, px/v py/v pz/v)

Question 29

How do you find the surface area of an open region?

Answer 29

intintS dS = intintD ||ru cross rv|| dA

Question 30

How do you evaluate a surface integral?

Answer 30

Let S be a surface with g(x,y) and let R be its projection onto the xy plant. intintS f(x,y,z) dS = intintR f(x,y,g(x,y))*sqrt(1 + gx(x,y)^2 + gy(x,y)^2) dA

Question 31

How do you find the flux integral?

Answer 31

Let F(x,y,z) (vector field) = Mi + Nj + Pk. intintS = F dot N dS = intintR F dot [-gx(x,y)i - gy(x,y)j + k] dA (Oriented upward), intintR F dot [gx(x,y) + gy(x,y) -k] dA (Oriented downward)