Section 5.4 Week 5 Activity
Subsection 5.4.1 Proofs for Differential Operators
Activity 5.4.1.
Prove the four statements in Proposition 5.3.8. You can work entirey with vector in \(\RR^3\text{.}\)
Each of the linearity properties involves expanding the vector field in components and using the linearity of single-variable derivatives in the components. I'll just show the divergence identity as an example. I expand the components and then apply the divergence operator. I use the rules for vector addition and scalar multiplication.
I use the linearity of the single variable derivatives.
Then I reoder the terms.
The other three solutions follow this pattern almost exactly, just with different use of the specific rules for vector algebra. The calculations for the cross product are a bit lengthy, but no more conceptually challenging than this.
Activity 5.4.2.
Prove this identity for vector fields in \(\RR^3\text{.}\)
I expand in components and apply definition of the cross product.
I use linearity and the power rule for the partial derivatives to expand this further.
I have twelve terms here. I reorder them in the following way.
From here, I just need to recognize that the result is precisely the expanded form of the right side that I'm trying to work towards.
Subsection 5.4.2 Vector Field Differential Operators
Activity 5.4.3.
Determine if the following vector field is irrotational and/or incompressible.
I need to calculate both the curl and the divergence.
The field is both irrotational and incompressible.
Activity 5.4.4.
Determine if the following vector field is irrotational and/or incompressible.
I need to calculate both the curl and the divergence.
The field is neither irrotational nor incompressible.
Activity 5.4.5.
Determine if the following vector field is irrotational and/or incompressible.
I need to calculate both the curl and the divergence.
The field neither irroational or incompressible
Activity 5.4.6.
Determine if the following vector field is irrotational and/or incompressible.
I need to calculate both the curl and the divergence.
The field is irrotational and incompressible.
Activity 5.4.7.
Determine if the following vector field is irrotational and/or incompressible.
I need to calculate both the curl and the divergence.
The field is irrotational but not incompressible.
Activity 5.4.8.
Determine if the following vector field is irrotational and/or incompressible.
I need to calculate both the curl and the divergence.
The field is neither irrotational nor incompressible.
Subsection 5.4.3 Harmonic Scalar Fields
Activity 5.4.9.
Determine if this scalar field is harmonic.
I calculate the Laplacian (the divergence of the gradient).
The field is not harmonic.
Activity 5.4.10.
Determine if this scalar field is harmonic.
I calculate the Laplacian (the divergence of the gradient).
The field is harmonic.
Activity 5.4.11.
Determine if this scalar field is harmonic.
I calculate the Laplacian (the divergence of the gradient).
The field is not harmonic.
Activity 5.4.12.
Determine if this scalar field is harmonic.
I calculate the Laplacian (the divergence of the gradient).
The field is harmonic.
Subsection 5.4.4 Conceptual Review Questions
What is a vector field?
What is an integral curve?
What is curl?
What is divergence?
How does the Leibniz rule extend?