calc 3 test
Anybody want a practice test in calc 3? (vector calc)
Math 2500 sp08 Test 4, 4/25/2008 NAME
Review of operator symbols: Dx means differentiation wrt x, so “multiplication by Dx” means differentiation wrt x. Thus: If f is a function, Dxf = partial derivative of f wrt x;
and if we define “del” = ∇ = (Dx, Dy, Dz), then ∇f = (Dxf, Dyf, Dzf) = grad(f); and
if F = (M,N,P) is a vector field, then ∇×F= (DyP-DzN, DzM-DxP, DxN-DyM) = curl(F); and ∇•F = DxM + DyN + DzP = div(F).
Recall also dxdy = (dx/ds dy/dt – dx/dt dy/ds) dsdt.
(15) IA. Important theorems:
a) If C is a smooth curve going from point p to q, and f is a smooth function on C, what does the fundamental theorem of one variable calculus give as the value of the path integral (i.e. “work” for a force field, “flow” for a velocity field) of F = ∇f, along C?
b) If C is the boundary curve of a smooth surface S, and F = (M,N,P) is a smooth vector field on S, state Stokes’ thm. relating the path integral of F along C, to a surface integral.
c) If S is the smooth boundary surface of a bounded region R in 3 - space, and F = (M,N,P) is a smooth vector field on R, state the divergence theorem relating the flux integral of F across S, to a volume integral.
(15) IB. Important facts: True or false? (and briefly why or why not)
a) If f is a smooth function in a region R in space, then curl(gradf) is always = 0 in R.
b) If F = (M,N,P) is a smooth vector field in a region R in space, curl(F) = 0 in R, and C is a closed curve in R, the path integral of Mdx +Ndy+Pdz along C is always zero.
c) If G is a smooth vector field in a region R in space with curl(G) = 0, and U is a simply connected subregion of R, there is a smooth function f in R, with gradf = G.
d) If F is a smooth vector field in space, defined on two smooth surfaces S,T having the same (oriented) boundary curve, the (flux) integral of ∇×F over S, or over T is the same.
e) If G is a smooth vector field defined in all of 3 space, and div(G) = 0 (everywhere), then the (flux) integral of G over the surface of any sphere is zero.
II.a) Let R be the plane region inside the ellipse C: (x/2)^2 + (y/3)^2 = 1.
If F = (0,x), the flow of F around C is computed by the path integral ∫C x dy.
Compute this integral using the parametrization x = 2cos(t), y = 3sin(t), 0 ≤ t ≤ 2pi.
IIb) If we apply Green’s theorem to the path integral above, what double integral does it equal, over R? Compute that double integral, changing variables by the parametrization x = 2s cos(t), y = 3s sin(t), for 0 ≤ s ≤ 1, 0 ≤ t ≤ 2pi, and the “recalled” formula for dxdy.
(You should get the same result. What geometric quantity have you computed?)
III. Let H be the hemisphere of radius 2, x^2 + y^2 + z^ 2 = 4, z ≥ 0, and
Define the vector field F = (xz, x + yz, y^2).
a) Compute ∇×F =
b) Show the flux of ∇×F outward through H equals 4pi, in one of these ways:
i) Explain, with minimal computing, why it equals the area of the circle x^2 + y^2 = 4.
ii) Compute it as a path integral using Stokes.
iii) (last resort) parametrize H and actually compute the flux integral.
IV. Let S be the boundary surface of the solid tetrahedron T with vertices (0,0,0), (1,0,0), (0,1,0), (0,0,1), and let F = (xz cos^2(z), yz sin^2(z), yx).
a) Compute div(F) = ∇•F = ?
b) Compute the flux of F outward through S either directly as a surface integral (masochists only) or by using the divergence theorem.
