vector calc test
heres a little non honors vector calc test.
2500 spring 2008 exam NAME.
I. A) True or false? and briefly why?
i) The path integral of any gradient field over any path is always zero.
ii) A vector field with curl equal to zero in any region is always a gradient field in that region.
iii) A vector field defined everywhere in three space, with divergence zero everywhere, has flux equal to zero through the boundary surface of any bounded region.
iv) The derivative of a function f in a given direction is obtained by dotting the gradient of f with the appropriate direction vector.
B) Give an explicit detailed computational answer:
i) If F = (M,N,P) is a vector field, then curl F = ?
ii) If F = (M,N,P) ) is a vector field, then div(F) = ?
iii) If f(x,y,z) is a smooth real valued function, then curl(grad(f)) = ?
II. Vectors. Given the three points A = (-1,0,4), B = (2,2,6), C = (1,2,1), find:
i) the vectors v = B-A = ; w = C-A =
ii) (dot product) v.w = ; (cross product) v x w
iii) cosine of the angle A in triangle ABC.
iv) area of triangle ABC.
v) equation of plane containing A,B,C.
vi) parametric equation of line normal to that plane, and through point A.
III. Vector valued functions.
A. If r(t) = (cos(t), sin(t), 2 t^(3/2)),
i) find the velocity vector to the corresponding curve at t = π.
ii) find the equation of the tangent line to this curve at t = π.
iii) Find the arclength of the curve over the interval 0 ≤ t ≤ 9.
IV. Derivatives of functions of several variables
A) Let f(x,y,z) = xy + yz^2 + xz^3.
i) Find the gradient vector for this f.
ii) Find the derivative of this f in the direction of the vector u = (-2/3, -1/3, 2/3),
at the point p = (2,0,3).
iii) Find the equation of the tangent plane to the level surface {f = 3}, for this f, at the point p = (1,1,1).
B) Now let g(x,y) = x^3 – 3xy + y^3.
i) Find all “critical points” of g (i.e. points where grad(g) = (0,0)).
ii) Test those critical points and identify as local max, local min, or saddle point.
V. LaGrange multipliers:
Use LaGrange’s method to find the maximum and minimum values, of the function f = x+y restricted to the ellipse x^2 + 2y^2 = 6.
VI. Double integrals:
Compute the y coordinate of the centroid of the plane triangle bounded by the x and y axes and the line x+y = 1.
VII.Triple integrals: The (truncated) cone z^2 = x^2 + y^2, for 0 ≤ z ≤ 1, can also be described in cylindrical coordinates as the cone z = r, for 0 ≤ z ≤ 1.
i) Find the volume of the region inside this (flat topped) cone by triple integration in cylindrical coordinates.
ii) Find the z coordinate of the centroid of the same cone by triple integration in cylindrical coordinates.
VIII. The cone z^2 = x^2 + y^2, z ≥ 0, can be described in spherical coordinates as φ = π/4; and the sphere of radius ½ centered at (0,0,1/2) as ρ = cos(φ), 0 ≤ φ ≤ π/2. Find the volume of the “ice cream cone” lying above the cone and inside the sphere, by triple integration in spherical coordinates.
IX. Circulation and Green’s theorem: Let F = (M,N) where M = x - [y/(x^2+y^2)], and
N = y + [x/(x^2+y^2)].
i) Parametrize the unit circle C by polar coordinates (counter clockwise) and compute the path integral (circulation) of F around C.
ii) Compute ∂M/∂y.
iii) Compute ∂N/∂x.
iv) Are your results in i),ii),iii), consistent with (compatible with) Greens theorem? Explain.
X. Flux and Divergence: Let F = (z,y,x), and parametrize the unit sphere by x = sin(φ)cos(θ),
y = sin(φ)sin(θ), z = cos(φ).
i) Compute (∂x/∂φ, ∂y/∂φ, ∂z/∂φ).
ii) Compute (∂x/∂θ, ∂y/∂θ, ∂z/∂θ).
iii) Compute the cross product of the two vectors in i) and ii).
iv) Write down, but do not compute, the parametrized surface integral for the flux of F through the unit sphere, as a double integral in the variables (φ,θ).
v) Compute div(F), and use the divergence theorem to calculate the flux in iv) as a familiar volume integral.
