Final

Question 1

if two vectors are orthogonal, then

Answer 1

a dot b = 0

Question 2

what is the scalar projection of b onto a

Answer 2

(a dot b) / (|| a ||)

Question 3

what is the vector projection of b onto a

Answer 3

( (a dot b) / (|| a ||)^2 ) * a

Question 4

how do you find a vector orthogonal to both a and b

Answer 4

a x b

Question 5

how do you tell if two vectors are parallel

Answer 5

a x b = 0

Question 6

how do you find a volume of three vectors

Answer 6

|| a dot (b x c) ||

Question 7

what is the parametric equation through the point (x0, y0, z0) and parallel to the point

Answer 7

x = x0 + at

Question 8

how do you find a symmetric equation of a line

Answer 8

find the parametric equation and solve for t

Question 9

what is the equation of the line segment from r0 to r1

Answer 9

r(t) = ( 1 - t )r0 + tr1

Question 10

what is the scalar equation of a plane through the point P(x0, y0, z0) with the normal vector n(a, b, c)

Answer 10

a(x-x0) + b(y-y0) + c(z-z0) = 0

Question 11

what is the distance from the point P(x0, y0, z0) to the plane ax + by + cz + d = 0

Answer 11

|| ax0 + by0 + cz0 + d || / sqrt( a^2 + b^2 + c^2 )

Question 12

what is an equation of a tangent plane to the surface z = f(x,y) at the point P(x0, y0, z0)

Answer 12

z - z0 = fx(x0,y0)(x - x0) + fy(x0,y0)(y - y0)

Question 13

what is the linearization of f at a, b

Answer 13

L(x,y) = f(a,b) + fx(a,b) (x - a) + fy(a,b) (y - b)

Question 14

explain the chain rule using the tree method

Answer 14

create a diagram that relates each of the variables and the partials to get to new variables. for example, z is in terms of x and y so (partial z) / (partial x) gives in terms of x, now x and y are in terms of s and t so (partial x/ partial t) gives t and (partial y / partial t) gives t so when added you get the equation with respects to t

Question 15

how do we use implicit differentiation in two variables

Answer 15

dy/dx = - (Fx)/(Fy)

Question 16

what is the directional derivative of a unit vector u = <a></a>

Answer 16

Du f(x,y) = fx(xy) a + fy(xy) b

Question 17

what is a directional derivative with three variables

Answer 17

Du f(x,y,z) = gradient f(x,y,z) dot u

Question 18

how do you find a local maximum or minimum

Answer 18

1. complete the second derivative test
D(a,b) = fxx(a,b) fyy(a,b) - [fxy(a,b)]^2
2. figure out results
D>0 & fxx>0 --> local min
D>0 & fxx<0 --> local max
D<0               --> saddle point

Question 19

how do you find critical points

Answer 19

find all points where the first derivative = 0 or does not exist

Question 20

how do you find absolute max and mins

Answer 20

1. find all critical points
2. find all extreme values of f on the boundary of D
3. the largest values and smallest values from steps 1 and two are the absolute max and min

Question 21

what is the method of Lagrange multipliers

Answer 21

this is a way of finding maximum and minimum values of f(x,y,z) subject to the constraint g(x,y,z) = k

1. find all values of (x,y,z) where
gradient f(x,y,z) = lambda gradient g(x,y,z)
and g(x,y,z) = k

2. evaluate all found points from part one to find the largest and smallest values

Question 22

how do you convert between polar and rectangular coordinates

Answer 22

r^2 = x^2 + y^2 
x = r cos theta
y = r sin theta

Question 23

how do you find surface area

Answer 23

double integral of sqrt( [fx(x,y)]^2 + [fy(xy)]^2 +1 )

Question 24

how to convert from rectangular to cylindrical coordinates

Answer 24

r^2 = x^2 + y^2
tan theta = y/x
z=z

Question 25

what is a triple integral of spherical coordinates

Answer 25

SSS ( [(roe)sin(phi)cos(theta), (roe)sin(phi)sin(theta), (roe)cos(phi)] (roe)^2sin(phi) d(roe)d(theta)d(phi)

Question 26

what is the line integral of F along C

Answer 26

S (F dot dr) = S [F(r(t)) dot r’(t)] dt = S (F dot T) ds

Question 27

what is the formula for a line integral

Answer 27

S (Pdx + Qdy + R) dz

Question 28

when is a vector field conservative

Answer 28

partial P / partial y = Partial Q / partial x

Question 29

what is greens theorem

Answer 29

SS ( (partial Q / Partial x) - (partial P / partial y))

Question 30

find the area of D using greens theorem

Answer 30

S(x)dy = -S(y)dx = 1/2 S ( [x dy] - [y dx]

Question 31

what is the curl of F

Answer 31

Gradient cross F

Question 32

what is the divergence of F

Answer 32

Gradient dot F

Question 33

what is a line integral using curl

Answer 33

SS (curl F) dot K dA