Chapter 9: Vector Calculus
Section 9.3: Differential Operators
Example 9.3.13
Obtain the Laplacian of the scalar function . Show that it is equivalent to the divergence of the gradient.
Solution
Mathematical Solution
In Cartesian coordinates, , the Laplacian of , is .
In Cartesian coordinates, , the gradient of , is the vector
The divergence of this vector is
Maple Solution - Interactive
Initialize
Tools≻Load Package: Student Vector Calculus
Loading Student:-VectorCalculus
Define the scalar field
Context Panel: Assign to a Name≻
Obtain , the Laplacian of
Common Symbols palette: Del operator Context Panel: Evaluate and Display Inline
=
Alternate implementation of the Laplacian
Write the name . Context Panel: Evaluate and Display Inline
Context Panel: Student Vector Calculus≻Differentiate≻Laplacian (Complete the dialog as per the figure below.)
Obtain the Laplacian as the divergence of the gradient
Common Symbols palette: Del and dot-product operators
Context Panel: Evaluate and Display Inline
Obtain the Laplacian from first principles
Calculus palette: partial-derivative operator Context Panel: Evaluate and Display Inline
Maple Solution - Coded
Load the Student VectorCalculus package.
Use , the assignment operator.
Obtain the Laplacian of
Apply the Laplacian command.
Obtain as , the divergence of the gradient of
To the result of the Gradient command, apply the Divergence command.
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