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14.4
Explain the process to find an eq. for the tangent plane.
Given: f(x,y), pt.
 1. Find the gradient vector.
 2. Plug the point into the
 3. Insert given pt. and gradient vector into the equation of a plane (where z=1).

14.6
Find the directional derivative.
Given: f(x,y), vector, and pt.
 1. Find the gradient vector
 2. Find the unit vector of the given pt.
 3. dot product of 1. and 2.

14.7
In which direction is the rate of change of f(x,y) maximized.
Given: gradient vector.
unit vector of the gradient vector

14.7
Find the max rate of change.
Given: gradient vector
magnitude of the gradient vector.

14.8
Find the max value of f(x,y) subjected to g(x,y).
 1. Find the gradient vector with respect to f(x,y) and g(x,y).
 2. Solve of λ.
 3. Solve for x or y.
 4. Substitute to find y or x.
 5. Find the max by substituting x and y into g(x,y).

12.5
Find and eq. for the plane that passes through a pt. and contains a parametric equation.
 1. Dot product of Q and R.
 2. QP X V.
 3. answer in eq. form.

14.4
Find the linearization of f(x,y) at pt.
 1. Find the gradient vector.
 2. Find z using pts in f(x,y).
 3. answer in eq. form

15.5
What is the mass m using the applications of double integrals?
p(x,y) dA; where the domain is D

15.5
What is the moment of the entire lamina about the xaxis using the applications of double integrals?
M _{x}= y * p(x,y) dA; where the domain is D

15.5
What is the moment of the entire lamina about the yaxis using the applications of double integrals?
My= x * p(x,y) dA; where the domain is D

15.5
What is the polar moment of inertia using the applications of double integrals?
I _{o}= (x ^{2}+y ^{2}) p(x,y) dA; where the domain is D

15.5
What is the moment of inertia of the lamina about the xaxis using the application of double integrals?
I _{x}= (y ^{2})p(x,y) dA; where the domain is D

15.5
What is the moment of inertia of the lamina about the yaxis using the application of double integrals?
I _{y}= (x ^{2}) p(x,y) dA; where the domain is D

15.9
What is the formula for triple integration in spherical coordinates?
 E= f(psinΦcosθ, psinΦsinθ, pcosΦ)p^{2}sinΦ dpdθdΦ;
 where E{(p,θ,Φ)a≤p≤b, α≤θ≤β, c≤Φ≤d}

15.9
Explain the triple integrals with spiracle coordinates using a diagram:

15.9
In spherical coordinates, what is x? y? z?
 x= psinΦcosθ
 y= psinΦsinθ
 z=pcosΦ

What is C?
 A smooth space curve given by the parametric equation:
 x=x(t) y=y(t) z=z(t)
or my a vecor:
r(t)=x(t) i +y(t) j +z(t) k

16.2
Evaluate a line integral, where C is the given curve.
Given: f(x,y,z),x,y,z, domain of t.
 1. f(x(t),y(t),z(t))√(dx/dt)^{2} +(dy/dt)^{2} +(dz/dt)^{2})dt
 ^{}

16.2
What is the representative of the line segment that starts at r_{0} and ends at r_{1?
}
r(t)=(1t)r_{0 }+ tr_{1}; where r_{0 }and r_{1 }are point on the line.

16.2
Evaluate the line integral F dr, where C is given by the vector functions r(t).
Given: F( x,y), r(t)
 1. Remember that r(t) goes into F.
 *F is the normal vector of the curve
 2. Find r'(t)
 3. Plug x and y of r into eq. F(r(x,y))
 4. Combine x's with x's and y's with y's
 5. use W= F(r(t)) ⋅ r^{'}(t)dt

16.3
How do you determine if the vector field F is a conservative function?
δP/δy =δQ/δx; throughout D

16.3
If F(x,y)= some i+ j, what are the steps to find a function f such that F=
 1. Find δP/δy =δQ/δx to be conservative.
 2. Find the partial with respect to x and y. i.e.<i,j>
 3. with respect to x.
 4. Take the partial with respect to y of 3..
 5. g^{'}(y) with respect to y.
 6 .Combine 3. and 5..

16.3
What is the fundamental theorem of line integrals?
F⋅d r = f(end)f(start)

16.4
What is the Green's Theorem?

?
"del" =

16.5
What is the curl?
;where = < i, j, k> and F= P,Q,R

16.5
curl( f)=
0


