MTH316 - Multivariable Calculus Assignment

Assignment Task

Question 1

a. The pressure acting on a triangular plate shown below has the function P(x,y).

If coordinates x and y are measured in meters, compute the net force acting on the triangular plate. With reference to the pressure acting over the plate, explain intuitively why the net force is positive (or negative).

(b) Define a solid region E as

E = {(x, y, z) | 1 < x>2 + Y²< 4>2}

(i) Sketch the projections of the region E onto the xy-plane and the yz-plane.

(ii) Set up a triple integral in cylindrical coordinates that gives the volume of solid E. Do not evaluate the integral.

(iii) Given that the density of solid E varies according to ρ(x,y) below, set up and evaluate a triple integral that gives the mass of solid E.

P(x,y) = 1/x2 + y2  (kg/m³)

Question 2

Define a force field (in Newtons) by

F(x,y) = [ x + 2 y/ 1 + x²y², 2x/1 + x²y²]^T

where coordinates x and y are in meters.

(a) Evaluate F(1, 0), F(0, -1), F(1, 1), F(-1, -1) and sketch them on the xy-plane.

(b) Show that the force field is conservative and hence evaluate a scalar potential function E(x,y) such that F = ∇E.

(c) A path C is parameterized by the function below. Calculate the work done by F(x, y) along the path C.

r(t) = [ t - 1 -1], 0 < t>

(d) With reference to the vector field, explain why the work done in part (c) is positive (or negative).

Question 3

A vector field is defined by F(x,y) and three paths C1, C2 and C3 are shown below.

(a) Evaluate directly the line integral of F(x, y) along the path C1 + C2.

(b) Evaluate directly the line integral of F(x, y) along the path C3.

(c) Use Green’s theorem to evaluate the line integral of F(x, y) along the path C1 + C2 + C3 and explain your answer in relation to parts (a) and (b).

Question 4

Suppose we want to best-fit a quadratic function y(x) = ax2 + b to three data points as shown below. This means we want to find the values of constants a and b such that the best-fit curve is ‘closest’ to the three points given.

(a) Given that the error, di, between each data point (xi, yi) and the best-fit curve is

di = y(x1) - yi = (axi2 + b) - yi

show that the sum of squared errors (SSE) is

SSE(a,b) = Σdi₂ = (b -1)² + (4a + b - 2) ² +(9a + b - 4)²

(b) The best-fit curve occurs when the SSE is at a minimum. Solve for the constants a and b of the best-fit curve and apply derivative tests to show that both the local & global minimum value of the SSE occurs at these constant values.

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