Chapter 9: Vector Calculus
Section 9.4: Differential Identities
Example 9.4.8
For sufficiently well-behaved scalars fx,y,z and gx,y,z, verify Identity 3 in Table 9.4.1.
Solution
Mathematical Solution
∇f g=∂x(f g)∂y(f g)∂z(f g)=f gx+g fxf gy+g fyf gz+g fz=f gxgygz+g fxfyfz=f ∇g+g ∇f
In the second column vector, the product rule of differentiation is applied to each component.
Maple Solution
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
Additional notational devices
The Suppress command in the Typesetting package allows suppression of arguments on input, as well as on output.
The declare command in the PDEtools package suppresses arguments on output, and sets partial derivatives as subscripts. Because the Suppress command acts first, the arguments can be suppressed in the ensuing declare command.
Typesetting:-Suppressfx,y,z,gx,y,z
PDEtools:-declaref,g,quiet
Implement Identity 3: ∇f g=f ∇g+g ∇f
Common Symbols palette: Del operator Context Panel: Evaluate and Display Inline
∇f g =
f ∇g+g ∇f =
On the left, use a space between the names f and g to indicate multiplication. On the right, after the function name f, again put a space before inserting the Del operator. Do the same after the name g and before the last Del operator.
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