Find curl and divergence of a vector field

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Matthew Mears

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Matthew Mears 7 years, 1 month ago

Created this as a copy of Find curl and divergence of a vector field.

Name	Status	Author	Last Modified
Find curl and divergence of a vector field	draft	Michael Proudman	07/12/2021 09:40
Find curl and divergence of a vector field	Needs to be tested	Simon Vaughan	13/01/2020 23:35
Find curl and divergence of a vector field	draft	Matthew Mears	28/03/2020 23:36
Ugur's copy of Find curl and divergence of a vector field	Ready to use	Ugur Efem	24/05/2024 11:24

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Question statement

For the vector field

$\boldsymbol{u}=\pmatrix{\simplify{{a1}*x^{p1}+{b1}*y^{p2}*z^{p3}},\simplify{{c1}*y^{p4}+{d1}*x^{p5}*z^{p6}},\simplify{{e1}*z^{p7}+{f1}*x^{p8}*y^{p9}}}$

calculate $\boldsymbol{\nabla}\times\boldsymbol{u}$ and $\boldsymbol{\nabla\cdot u}$ , and determine whether $\boldsymbol{u}$ is irrotational or solenoidal, or both.

Give any introductory information the student needs.

			Name	Type	Generated Value

f1	integer	0
irrequal	string	is equal
isirr	string	is irrotational
b1	integer	0
d1	integer	0
issol	string	is not solenoidal
e1	integer	6
irrotational	string	Yes
a1	integer	7
c1	integer	2
solenoidal	string	No
p2	integer	6
p3	integer	6
solequal	string	is not equal
p1	integer	9
p6	integer	1
p7	integer	2
p4	integer	2
p5	integer	4
notsolenoidal	string	Yes
p8	integer	9
p9	integer	4
notirrotational	string	No
n	integer	1

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Value

xxxxxxxxxx
 
if(n=1,0,random(1..9))

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Description

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Generated value: `integer`

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Parts

Part a) Gap-fill
1. Gap Mathematical expression
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Part b) Gap-fill
Part c) Choose one from a list
Part d) Choose one from a list

Gap-fill

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$\boldsymbol{\nabla\cdot u}=$ Unnamed gap.

Advice

The curl of a vector field $\boldsymbol{u}=\pmatrix{u_x,u_y,u_z}$ is given by

$\boldsymbol{\nabla}\times\boldsymbol{u}=\pmatrix{\frac{\partial u_z}{\partial y}-\frac{\partial u_y}{\partial z},\frac{\partial u_x}{\partial z}-\frac{\partial u_z}{\partial x},\frac{\partial u_y}{\partial x}-\frac{\partial u_x}{\partial y}}.$

The divergence of the same vector field is given by

$\boldsymbol{\nabla\cdot u}=\frac{\partial u_x}{\partial x}+\frac{\partial u_y}{\partial y}+\frac{\partial u_z}{\partial z}.$

a)

By partial differentiation

$\boldsymbol{\nabla \times u}=\pmatrix{\simplify{{f1*p9}*x^{p8}*y^{p9-1}+{-d1*p6}*x^{p5}*z^{p6-1}},\simplify{{b1*p3}*y^{p2}*z^{p3 -1}+{-f1*p8}*x^{p8-1}*y^{p9}},\simplify{{d1*p5}*x^{p5-1}*z^{p6}+{-b1*p2}*y^{p2-1}*z^{p3}}}.$

b)

Again, by partial differentiation

$\boldsymbol{\nabla\cdot u}=\simplify{{a1*p1}*x^{p1-1}+{c1*p4}*y^{p4-1}+{e1*p7}*z^{p7-1}}.$

A vector field is irrotational if its curl is equal to the zero vector; a vector field is solenoidal if its divergence is equal to zero.

c)

Since $\boldsymbol{\nabla}\times\boldsymbol{u}$ {irrequal} to the zero vector, the vector field {isirr}.

d)

Since $\boldsymbol{\nabla\cdot u}$ {solequal} to zero, the vector field {issol}.

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This question is used in the following exam:

End of term Review by Fouzia Ouali in Fouzia's workspace.

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Matthew Mears 7 years, 1 month ago

Error in variable testing condition

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Generated value: `integer`

Parts

Gap-fill

a)

b)

c)

d)

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Find curl and divergence of a vector field

Metadata

Taxonomy: mathcentre

Taxonomy: Kind of activity

Taxonomy: Context

Contributors

History

Comment

Checkpoint description

Matthew Mears 7 years, 1 month ago

Error in variable testing condition

Error

Warning

Generated value: integer

Parts

Gap-fill

a)

b)

c)

d)

Generated value: `integer`