Final Flashcards
if two vectors are orthogonal, then
a dot b = 0
what is the scalar projection of b onto a
(a dot b) / (|| a ||)
what is the vector projection of b onto a
( (a dot b) / (|| a ||)^2 ) * a
how do you find a vector orthogonal to both a and b
a x b
how do you tell if two vectors are parallel
a x b = 0
how do you find a volume of three vectors
|| a dot (b x c) ||
what is the parametric equation through the point (x0, y0, z0) and parallel to the point
x = x0 + at
how do you find a symmetric equation of a line
find the parametric equation and solve for t
what is the equation of the line segment from r0 to r1
r(t) = ( 1 - t )r0 + tr1
what is the scalar equation of a plane through the point P(x0, y0, z0) with the normal vector n(a, b, c)
a(x-x0) + b(y-y0) + c(z-z0) = 0
what is the distance from the point P(x0, y0, z0) to the plane ax + by + cz + d = 0
|| ax0 + by0 + cz0 + d || / sqrt( a^2 + b^2 + c^2 )
what is an equation of a tangent plane to the surface z = f(x,y) at the point P(x0, y0, z0)
z - z0 = fx(x0,y0)(x - x0) + fy(x0,y0)(y - y0)
what is the linearization of f at a, b
L(x,y) = f(a,b) + fx(a,b) (x - a) + fy(a,b) (y - b)
explain the chain rule using the tree method
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
how do we use implicit differentiation in two variables
dy/dx = - (Fx)/(Fy)
what is the directional derivative of a unit vector u = <a></a>
Du f(x,y) = fx(xy) a + fy(xy) b
what is a directional derivative with three variables
Du f(x,y,z) = gradient f(x,y,z) dot u
how do you find a local maximum or minimum
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
how do you find critical points
find all points where the first derivative = 0 or does not exist
how do you find absolute max and mins
- find all critical points
- find all extreme values of f on the boundary of D
- the largest values and smallest values from steps 1 and two are the absolute max and min
what is the method of Lagrange multipliers
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
- evaluate all found points from part one to find the largest and smallest values
how do you convert between polar and rectangular coordinates
r^2 = x^2 + y^2 x = r cos theta y = r sin theta
how do you find surface area
double integral of sqrt( [fx(x,y)]^2 + [fy(xy)]^2 +1 )
how to convert from rectangular to cylindrical coordinates
r^2 = x^2 + y^2
tan theta = y/x
z=z
what is a triple integral of spherical coordinates
SSS ( [(roe)sin(phi)cos(theta), (roe)sin(phi)sin(theta), (roe)cos(phi)] (roe)^2sin(phi) d(roe)d(theta)d(phi)
what is the line integral of F along C
S (F dot dr) = S [F(r(t)) dot r’(t)] dt = S (F dot T) ds
what is the formula for a line integral
S (Pdx + Qdy + R) dz
when is a vector field conservative
partial P / partial y = Partial Q / partial x
what is greens theorem
SS ( (partial Q / Partial x) - (partial P / partial y))
find the area of D using greens theorem
S(x)dy = -S(y)dx = 1/2 S ( [x dy] - [y dx]
what is the curl of F
Gradient cross F
what is the divergence of F
Gradient dot F
what is a line integral using curl
SS (curl F) dot K dA
