Lectures 1-6
Given two vectors, vector a <3, -1, 4> and vector b <1, 5, -2>, calculate the vector projection of vector a onto vector b.
What is < -⅓, -5/3, ⅔>?
This parametrized equation describes the intersection between the surface y=5x^3 + 7 and 2x - y + 3z = 6
What is r(t) = <t, 5t^3 + 7, (13 - 2t + 5t^3) / 3>?
What must the vector field satisfy in order to use Fundamental Theorem of Line Integrals?
What is conservative?
This is what div(curl F) is always equal to
What is 0?
This is the unique highest value of a function on its domain
What is strict global maximum?
Two vectors u and v have magnitudes |u| = 6 and |v| = 8. If their dot product u dot v = 24√3 what is the angle θ between the two vectors?
What is 30 degrees?
This equation represents the value of the partial derivative fxxyxy when given f(x,y) = cos(x2)5y + yesin(1/x)
What is fxxyxy = 0?
This is the curl of <ex, y2, ln(sin z)>
What is 0?
Find the minimum distance from the origin to the constraint g: x + 2y - z = 4.
(Hence, function to minimize should be f(x, y, z) = x^2 + y^2 + z^2)
What is 2√6 /3?
Determine the volume of the parallelepiped formed by the vectors p (1,0,1), q (2,3,0), and v (0,3,-4)
What is 6u3?
Let A be the part of the paraboloid z = x^2 + y^2 that lies within x^2 + 4y^2 = 9, Parametrize A
What is r(u,v)= <u,v,u2+v2>, u2+4v2<9
If S is the top and Q the bottom half of a unit sphere with outword pointing normals, and ∫∫s curl(F) =17, this would be the value of ∫∫Q curl(F)
What is -17?
Find and classify the critical points of: f(x, y) = 3x^2 - 2y^3 + 6xy + 1
What is (0,0) saddle point, (1, -1) local minimum?
Find the parametric equations of the line L that passes through the point P (1, -2, 3) and is parallel to the vector d = (4, 0, -1)
X = 1 + 4t, Y = -2, Z = 3 - t
Compute the volume of the solid that is under the paraboloid z = 9 - x2 - y2 and above the xy-plane.
What is 81pi/2?
This is the upward pointing normals of 2=x-5y+3z^2 at z=5
What is <1,-5, 30>?
Find the shortest distance between vectors a (1,2,3) + t(1,0,-1) and b (-1,0,1) + t(2,1,1)
What is (2√11)/11?
These are all the conditions of a curve necessary to apply Green’s Theorem
What is a counterclockwise, smooth, simple closed curve?