Mathematical and Numerical Methods Assignment -

Learning Outcomes Assessed: This assessment is designed to assess your ability in the following one module learning outcome (copied from the "Module Description" appendix in the Module Guide): Perform vector algebra and calculus, including evaluations of gradient, divergence and curl and applications of (integral) theorems linking these quantities.

Question 1 - If φ = xze^{y^2} and F = x²yi - zcos(y)j + ye^zk

Show that

curl(φ, F) = φ.curl(F) - F x grad(φ).

Question 2 - Le the curve C be defined by

r(t) = e^tcos(t)I + e^tsin(t)j + 2k.

Find the tangent line and normal plane to C at the point at which t = π/4.

Question 3 - Find the normal line and tangent plane to the surface

9x² - 4y² - 25z² - 40 = 0

at the point (4, 1, -2).

Question 4 - Find the value of s for which the function

F = sz²i + (3y²z + 1)j + (y³ - 2xz)k

Satisfies curl(F) = 0.

For this value of s, find a function φ such that

F = grad φ.