Chapter 9: Vector Calculus
Section 9.3: Differential Operators
Example 9.3.13
Obtain the Laplacian of the scalar function fx,y=x/y+y/x. Show that it is equivalent to the divergence of the gradient.
Solution
Mathematical Solution
In Cartesian coordinates, ∇2f, the Laplacian of f, is ∂2f∂x2+∂2f∂y2=2 yx3+2 xy3.
In Cartesian coordinates, ∇f, the gradient of f, is the vector
fx i+fy j=1y−yx2 i+2x−xy2 j
The divergence of this vector is
∂x1y−yx2+∂y 2x−xy2=2 yx3+2 xy3
Maple Solution - Interactive
Initialize
Tools≻Load Package: Student Vector Calculus
Loading Student:-VectorCalculus
Define the scalar field f
Context Panel: Assign to a Name≻f
x/y+y/x→assign to a namef
Obtain ∇2f, the Laplacian of f
Common Symbols palette: Del operator Context Panel: Evaluate and Display Inline
∇2f = 2yx3+2xy3
Alternate implementation of the Laplacian
Write the name f. Context Panel: Evaluate and Display Inline
Context Panel: Student Vector Calculus≻Differentiate≻Laplacian (Complete the dialog as per the figure below.)
f = xy+yx→Laplacian2yx3+2xy3
Obtain the Laplacian as the divergence of the gradient
Common Symbols palette: Del and dot-product operators
Context Panel: Evaluate and Display Inline
∇·∇f = 2yx3+2xy3
Obtain the Laplacian from first principles
Calculus palette: partial-derivative operator Context Panel: Evaluate and Display Inline
∂2∂x2 f+∂2∂y2 f = 2yx3+2xy3
Maple Solution - Coded
Load the Student VectorCalculus package.
withStudent:-VectorCalculus:
Use ≔, the assignment operator.
f≔x/y+y/x:
Obtain the Laplacian of f
Apply the Laplacian command.
Laplacianf = 2yx3+2xy3
Obtain ∇2f as ∇·∇f, the divergence of the gradient of f
To the result of the Gradient command, apply the Divergence command.
DivergenceGradientf = 2yx3+2xy3
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