Chapter 9: Vector Calculus
Section 9.4: Differential Identities
Working with sufficiently well-behaved quantities in polar coordinates, verify Identity 5 in Table 9.4.1 for fr,θ and F=ur,θ er+vr,θ eθ.
Identity 5: ∇·f F= f ∇·F+F·∇f
The left side is the divergence of the product f F, where f is a scalar and F is a vector, both in polar coordinates. Hence, the resulting scalar is
f ur+∂rf u+∂θf vr = f ur+fr u+f ur+fθ v+f vθr
The first term on the right is the product of the scalar f and the divergence of F. The resulting scalar is
fur+ur+vθr=f ur+f ur+f vθr
The second term on the right is the dot product of F with the gradient of f. The resulting scalar is
uv·frfθr = u fr+v fθr
The sum of the two terms on the right is then
f ur+f ur+f vθr+u fr+v fθr
=f ur+fr u+f ur+fθ v+f vθr
Tools≻Load Package: Student Vector Calculus
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.
Define the vector field F
Write the free vector whose components are those of F.
Context Panel: Evaluate and Display Inline
Context Panel: Student Vector Calculus≻Conversions≻Apply Co-ordinate System≻
Context Panel: Student Vector Calculus≻Conversions≻To Vector Field
Context Panel: Assign to a Name≻F
u,v = u⁡r,θv⁡r,θ→apply coordinatesu⁡r,θv⁡r,θ→to Vector Fieldu⁡r,θv⁡r,θ→assign to a nameF
Implement Identity 5: ∇·f F=f ∇·F+F·∇f
Common Symbols palette: Del and dot-product operators
Gradient command from the Student VectorCalculus package
Press the Enter key.
Context Panel: Expand≻Expand
On the right, explicit use must be made of the Gradient command because f, a scalar, does not carry its coordinate system as an attribute. In typeset notation, the Del operator has no way of knowing which coordinate system to use when operating on a scalar. The alternative to invoking the Gradient command would be to set the ambient coordinate system to polar coordinates with the SetCoordinates command, an approach this Study Guide consistently avoids implementing.
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