1. Compute the divergence and curl of each of the following vector fields:
a) hx
2 + y
2
, x2 − y
2
, z2
i
b) hx + y, x − y, zi
2. For any two vector fields F, G show that:
∇.(F × G) = G.(∇ × F) − F.(∇ × G).
3. Given scalar functions u(x, y, z), v(x, y, z), w(x, y, z), φ(x, y, z) and a vector field F(x, y, z) =
hu(x, y, z), v(x, y, z), w(x, y, z)i, prove that
div(φF) = φ(div F) + (∇φ) · F.
4. Find a function f such that F = ∇f where F = hyz, xz, xy + 2zi
5. Let r = hx, y, zi and r = |r|, verify that
a) ∇ · r = 3
b) div (r
n
r) = (n + 3)r
n
6. Given u = xy2
z
2 and v = yz − 3x
2
, find ∇ · [(∇u) × (∇v)]
7. Given the vector field F = he
x
sin y, ex
cos y, zi, show that F is irrotational, hence or
otherwise, find the functions f such that ∇f = F.
The answer to the question is available in the PDF file https://www.assignmentexpert.com/https://www.assignmentexpert.com/homework-answers/mathematics-answer-72164.pdf
Dear visitor, please use panel for submitting new questions
Lastone
04.01.18, 09:22
6. Find the absolute minimum of f(x, y) = x 2 + y 2 + 2y − 1 on D =
{(x, y)|x 2 + y 2 4 ≤ 1} 7. Use Lagrange multipliers to find the
maximum and minimum values of the function subject to the given
constraint(s). a) f(x, y, z) = xyz; x 2 + 2y 2 + 3z 2 = 6 b) f(x, y,
z) = yz + xy; xy = 1, y2 + z 2 = 1 8. Find the first four terms in the
Taylor series expansion of the function: f(x, y) = e −y 2 cos(x + y)
about the point (0, 0).
Leave a comment
Thank you! Your comments have been successfully added. However, they need to be checked by the moderator before being published.
Numbers and figures are an essential part of our world, necessary for almost everything we do every day. As important…
APPROVED BY CLIENTS
"assignmentexpert.com" is professional group of people in Math subjects! They did assignments in very high level of mathematical modelling in the best quality. Thanks a lot
Comments
Dear visitor, please use panel for submitting new questions
6. Find the absolute minimum of f(x, y) = x 2 + y 2 + 2y − 1 on D = {(x, y)|x 2 + y 2 4 ≤ 1} 7. Use Lagrange multipliers to find the maximum and minimum values of the function subject to the given constraint(s). a) f(x, y, z) = xyz; x 2 + 2y 2 + 3z 2 = 6 b) f(x, y, z) = yz + xy; xy = 1, y2 + z 2 = 1 8. Find the first four terms in the Taylor series expansion of the function: f(x, y) = e −y 2 cos(x + y) about the point (0, 0).
Leave a comment