Vector Calculus (MAST20009)

the volume of the solid bounded by the graph z = f (x, y) = 1 + 2x + 3y, the rectangle R = [1, 2] × [0, 1] and the vertical sides of R;

g(x,y) =  ye^(−1/x^2), for (x,y) with x ̸= 0, and g(x,y)=y for (x,y) with x=0

Calculate lim g(x,y)→(0,0)

Define oblate spheroidal coordinates (u, θ, φ) by
x = coshucosθcosφ, y = coshucosθsinφ, z=sinhusinθ

where u≥0,−π2<θ<π2, 0≤φ<2π.

find an expression for ∇(u^2θ^3 + φ^4), in terms of u,  θ, φ

Let T be the unit tangent vector, N be the unit principal normal vector and B be the

unit binormal vector to a C3 path. Prove the Frenet-Serret formula dN/ds = τ B − κT
where κ is the curvature of the path and τ is the torsion of the path.

Hint: Differentiate B × T with respect to arclength.

Let P be the parallelgram in the xy-plane with vertices
A = (0, 0), B = (2, 1), C = (3, −2) and D = (1, −3).
Let P ′ be the square in the uv-plane with vertices
A′ = (0, 0), B′ = (1, 0), C′ = (1, 1) and D′ = (0, 1).
(a) Find a bijective linear map that maps P ′ to P such that A′ (B′, C′, D′ resp.) is mapped
to A (B, C, D resp.).
Page 6 of 11 pages

Let (x, y, z) ∈ R3 be parametrised by
T(u1, u2, u3) = (3u1 cos(u2), 4u1 sin(u2), 5u3)
for u1 ≥ 0, 0 ≤ u2 ≤ 2π and u3 ∈ R. Then the unit tangent vectors along the u1-curves,
u2-curves, u3-curves give a curvilinear coordinate system in R3. Find the unit tangent vectors
at each point (x, y, z) = T(u1, u2, u3) ∈ R3. Determine if this curvilinear system is orthogonal
or not. Explain your answer