MA5000 Mathematical And Numerical Methods Assignment,

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 ? = xzey^2 and F = x2yi - zcos(y)j + yezk

Show that

curl(?, F) = ?.curl(F) - F x grad(?).

Question 2 - Le the curve C be defined by

r(t) = etcos(t)I + etsin(t)j + 2k.

Find the tangent line and normal plane to C at the point at which t = ?/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 ? such that

F = grad ?.