Find gradient of scalar field

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Bill Foster

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Has some problems

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Bill Foster 6 years, 6 months ago

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Christian Lawson-Perfect 7 years, 9 months ago

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Christian Lawson-Perfect 7 years, 9 months ago

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Name	Status	Author	Last Modified
Find gradient of scalar field	Has some problems	Christian Lawson-Perfect	20/11/2019 14:51
Harry's copy of Find gradient of scalar field	draft	Harry Flynn	09/03/2018 14:13
Harry's copy of Find gradient of scalar field	draft	Harry Flynn	09/03/2018 14:15

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Question statement

Find $\boldsymbol{\nabla}f$ for the following function $f(x,y,z)$ .

Give any introductory information the student needs.

			Name	Type	Generated Value

p2	integer	2
p3	integer	4
p1	integer	1
p6	integer	4
p7	integer	0
p4	integer	3
p5	integer	4
p8	integer	0
p9	integer	1
a1	number	-5
p11	integer	0
p12	integer	1
t	integer	0
b1	number	1
p10	integer	1
c1	number	4
e1	number	-1
d1	number	3

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$f(x,y,z)=\simplify[std]{{a1}*x^{p1}*y^{p2}*z^{p3}+{b1}*x^{p4}*y^{p5}*z^{p6}+{c1}*({1-t}*sin({d1}*x^{p7}*y^{p8}*z^{p9})+{t}*cos({e1}*x^{p10}*y^{p11}*z^{p12}))}$ .

$\boldsymbol{\nabla}f=($ Gap 0. $,$ Gap 1. $,$ Gap 2. $)$ .

Advice

This question is simply an exercise in partial differentiation, using the fact that

$\boldsymbol{\nabla}f=\pmatrix{\frac{\partial f}{\partial x},\frac{\partial f}{\partial y},\frac{\partial f}{\partial z}}.$

The partial derivatives of $f$ are as follows:

$\begin{align} \frac{\partial f}{\partial x} &= \simplify{{p1*a1}*x^{p1-1}*y^{p2}*z^{p3}+{p4*b1}*x^{p4-1}*y^{p5}*z^{p6}+{c1*d1*p7*(1-t)}*x^{p7-1}*y^{p8}*z^{p9}*cos({d1}*x^{p7}*y^{p8}*z^{p9})-{c1*e1*p10*t}*x^{p10-1}*y^{p11}*z^{p12}*sin({e1}*x^{p10}*y^{p11}*z^{p12})} \\[0.5em] \frac{\partial f}{\partial y} &= \simplify{{p2*a1}*x^{p1}*y^{p2-1}*z^{p3}+{p5*b1}*x^{p4}*y^{p5-1}*z^{p6}+{c1*d1*p8*(1-t)}*x^{p7}*y^{p8-1}*z^{p9}*cos({d1}*x^{p7}*y^{p8}*z^{p9})-{c1*e1*p11*t}*x^{p10}*y^{p11-1}*z^{p12}*sin({e1}*x^{p10}*y^{p11}*z^{p12})} \\[0.5em] \frac{\partial f}{\partial z} &= \simplify{{p3*a1}*x^{p1}*y^{p2}*z^{p3-1}+{p6*b1}*x^{p4}*y^{p5}*z^{p6-1}+{c1*d1*p9*(1-t)}*x^{p7}*y^{p8}*z^{p9-1}*cos({d1}*x^{p7}*y^{p8}*z^{p9})-{c1*e1*p12*t}*x^{p10}*y^{p11}*z^{p12-1}*sin({e1}*x^{p10}*y^{p11}*z^{p12})} \end{align}$

Hence

$\boldsymbol{\nabla}f = \pmatrix{ \simplify{{p1*a1}*x^{p1-1}*y^{p2}*z^{p3}+{p4*b1}*x^{p4-1}*y^{p5}*z^{p6}+{c1*d1*p7*(1-t)}*x^{p7-1}*y^{p8}*z^{p9}*cos({d1}*x^{p7}*y^{p8}*z^{p9})-{c1*e1*p10*t}*x^{p10-1}*y^{p11}*z^{p12}*sin({e1}*x^{p10}*y^{p11}*z^{p12})}, & \simplify{{p2*a1}*x^{p1}*y^{p2-1}*z^{p3}+{p5*b1}*x^{p4}*y^{p5-1}*z^{p6}+{c1*d1*p8*(1-t)}*x^{p7}*y^{p8-1}*z^{p9}*cos({d1}*x^{p7}*y^{p8}*z^{p9})-{c1*e1*p11*t}*x^{p10}*y^{p11-1}*z^{p12}*sin({e1}*x^{p10}*y^{p11}*z^{p12})}, & \simplify{{p3*a1}*x^{p1}*y^{p2}*z^{p3-1}+{p6*b1}*x^{p4}*y^{p5}*z^{p6-1}+{c1*d1*p9*(1-t)}*x^{p7}*y^{p8}*z^{p9-1}*cos({d1}*x^{p7}*y^{p8}*z^{p9})-{c1*e1*p12*t}*x^{p10}*y^{p11}*z^{p12-1}*sin({e1}*x^{p10}*y^{p11}*z^{p12})} }.$

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Bill Foster 6 years, 6 months ago

Christian Lawson-Perfect 7 years, 9 months ago

Christian Lawson-Perfect 7 years, 9 months ago

Error in variable testing condition

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Generated value: `integer`

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Find gradient of scalar field

Metadata

Taxonomy: mathcentre

Taxonomy: Kind of activity

Taxonomy: Context

Contributors

Feedback

History

Comment

Checkpoint description

Bill Foster 6 years, 6 months ago

Christian Lawson-Perfect 7 years, 9 months ago

Christian Lawson-Perfect 7 years, 9 months ago

Error in variable testing condition

Error

Warning

Generated value: integer

Parts

Gap-fill

Generated value: `integer`