Full answer: True or False In Exercises 7982, determine whether the statement is true or

Chapter 15, Problem 77

(choose chapter or problem)

In parts (a)-(h), prove the property for vector fields F and G and scalar function f. (Assume that the required partial derivatives are continuous.)

(a) curl \((\mathbf{F}+\mathbf{G})\) = curl  \(\mathbf{F}\) + curl \(\mathbf{G}\)

(b) curl \((\nabla f)=\nabla \times(\nabla f)=\mathbf{0}\)

(c) div \((\mathbf{F}+\mathbf{G})\) = div \(\mathbf{F}\) + div \(\mathbf{G}\)

(d) div \((\mathbf{F} \times \mathbf{G})\) = (curl \(\mathbf{F}) \cdot \mathbf{G}-\mathbf{F} \cdot\)(curl \(\mathbf{G})\)

(e) \(\nabla \times[\nabla f+(\nabla \times \mathbf{F})]=\nabla \times(\nabla \times \mathbf{F})\)

(f) \(\nabla \times(f \mathbf{F})=f(\nabla \times \mathbf{F})+(\nabla f) \times \mathbf{F}\)

(g) div \((f \mathbf{F})=f\) div \(\mathbf{F}+\nabla f \cdot \mathbf{F}\)

(h) div(curl \(\mathbf{F})=0\) (Theorem 15.3)

Text Transcription:

F + G = curl F + curl G

(nabla f) = nabla x (nabla f) = 0

(F + G) = div F + div G

(F x G)  = (curl F) cdot G - F cdot (curl  G)

nabla x [nabla f + (nabla x F)] = nabla x (nabla times F)

nabla x (f F}) = f(nabla x F) + (nabla f) x F

(f F) = f div F + nabla f cdot F

(curl F) = 0