Test 4: 14.7 to 15.6 Flashcards
In spherical coordinates, what is x equal to?
p sin phi cos theta
In spherical coordinates, what is y equal to?
p sin phi sin theta
In spherical coordinates, what is z equal to?
p cos phi
In spherical coordinates what is p^2 equal to?
x^2 + y^2 + z^2
In spherical coordinates what is tan theta equal to?
y/x
In spherical coordinates, what is phi equal to (rect)?
arccos(z / sqrt(x^2+y^2+z^2))
In spherical coordinates what is r^2 equal to?
p^2 sin^2 phi
in spherical coordinates what is p equal to?
sqrt(r^2+z^2)
In spherical coordinates what is phi equal to (cyl)?
arccos(z/sqrt(r^2+z^2))
What is the integral template and order of integration for cylindrical coordinates?
intintint (Q) f(x,y,z) = intintint r dz dr dtheta
What is the integral template and order of integration for spherical coordinates?
intintint (q) f(x,y,z) = intintint p^2 sin phi dp dphi dtheta
How do you find the Jacobian?
It is the determinant of the partials of x,y (rows) with respect to (u,v) - px/pupy/pv - py/pupx/pv
How do you find the change of variables for a Jacobian?
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.
How do you determine the u/v coordinates for a Jacobian?
Use the initial 4 equations and set u and v equal to their respective results.
How do you find the integral for a Jacobian?
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.
How do you determine if a vector field is conservative?
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.
How do you find the potential function for a vector field?
Integrate M dx, N dy, and P dz and combine the results, omitting duplicate expressions. Add +C.
How do you find the curl of a vector field?
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.
How do you find the divergence of a vector field?
Find the del operator dot the vector field’s MNP: pM/x + pN/y + pP/z. Plug in point.
How do you find a line integral of a non-vector field using ds format?
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.
For parameterizations given by r(t) = x(t)i + y(t)j + z(t)k, what is ds?
||r’(t)|| dt = sqrt(x’(t)^2 + y’(t)^2 + z’(t)^2).
How do you find a line integral of a vector field using dr format?
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.
How do you write F dot dr in differential form?
intC M dx + N dy + P dz. Replace x, y and z with the appropriate parameterization using t.
How do you use the fundamental theorem of line integrals?
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.
What is Green’s Theorem?
COUNTERCLOCKWISE. intC M dz + N dy = intintR (pN/x - PM/y) dA
What is the formula for the line integral for area?
A = 1/2 intC x dy - y dx
What is the equation for a parametric surface?
r(u,v) = x(u,v)i + y(u,v)j + z(u,v)k
How do you find the normal vector to a parametric surface?
By finding ru cross rv (ijk, px/u py/u pz/u, px/v py/v pz/v)
How do you find the surface area of an open region?
intintS dS = intintD ||ru cross rv|| dA
How do you evaluate a surface integral?
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
How do you find the flux integral?
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)
