Chapter 9: Vector Calculus
Section 9.4: Differential Identities
Example 9.4.2
If f=x y2z3, show that ∇f is curl-free.
Solution
Mathematical Solution
The gradient of f=x y2z3 is the vector ∇f=y2z32xyz33xy2z2 and its curl is the vector
∇×∇f=|ijk∂x∂y∂zy2z32 x y z33 x y2z2| = ∂y(3 x y2z2)−∂z(2 x y z3)∂z(y2z3)−∂x(3 x y2z2)∂x(2 x y z3)−∂y(y2z3) = 6 x y z2−6 x y z23 y2z2−3 y2z22 y z3−2 y z3 = 000
Maple Solution - Interactive
Initialize
Tools≻Load Package: Student Vector Calculus
Loading Student:-VectorCalculus
Tools≻Tasks≻Browse: Calculus - Vector≻ Vector Algebra and Settings≻ Display Format for Vectors
Press the Access Settings button and select "Display as Column Vector"
Display Format for Vectors
Obtain the curl of the gradient of f
Common Symbols palette: Del and cross-product operators
Context Panel: Evaluate and Display Inline
∇×∇x y2z3 =
Maple Solution - Coded
Load the Student VectorCalculus package and execute the BasisFormat command.
withStudent:-VectorCalculus:BasisFormatfalse:
Compute the curl of the gradient of f
Apply the Gradient and Curl commands from the Student VectorCalculus package.
CurlGradientx y2z3 =
<< Previous Example Section 9.4 Next Example >>
© Maplesoft, a division of Waterloo Maple Inc., 2024. All rights reserved. This product is protected by copyright and distributed under licenses restricting its use, copying, distribution, and decompilation.
For more information on Maplesoft products and services, visit www.maplesoft.com
Download Help Document
