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Chapter 14: Problem 65

Prove the following properties of the divergence and curl. Assume \(\mathbf{F}\) and \(\mathbf{G}\) are differentiable vector fields and \(c\) is a real number. a. \(\nabla \cdot(\mathbf{F}+\mathbf{G})=\nabla \cdot \mathbf{F}+\nabla \cdot \mathbf{G}\) b. \(\nabla \times(\mathbf{F}+\mathbf{G})=(\nabla \times \mathbf{F})+(\nabla \times \mathbf{G})\) c. \(\nabla \cdot(c \mathbf{F})=c(\nabla \cdot \mathbf{F})\) d. \(\nabla \times(c \mathbf{F})=c(\nabla \times \mathbf{F})\)

Short Answer

Expert verified
Question: Prove the following properties of divergence and curl: a) \(\nabla \cdot (\mathbf{F}+\mathbf{G}) = \nabla \cdot \mathbf{F} + \nabla \cdot \mathbf{G}\) b) \(\nabla \times(\mathbf{F}+\mathbf{G}) = \nabla \times \mathbf{F}+\nabla \times\mathbf{G}\) c) \(\nabla \cdot(c \mathbf{F}) = c(\nabla \cdot \mathbf{F})\) d) \(\nabla \times(c \mathbf{F}) = c(\nabla \times \mathbf{F})\) where \(\mathbf{F}\) and \(\mathbf{G}\) are differentiable vector fields, and \(c\) is a constant.

Step by step solution

01

Property a: Divergence of the Sum

To prove this property, we will use the definition of divergence: \(\nabla \cdot(\mathbf{F}+\mathbf{G})=\frac{\partial(F_x+G_x)}{\partial x}+\frac{\partial(F_y+G_y)}{\partial y}+\frac{\partial(F_z+G_z)}{\partial z}\) Now, we will apply the properties of the partial derivative: \(\nabla \cdot(\mathbf{F}+\mathbf{G})=\left(\frac{\partial F_x}{\partial x}+\frac{\partial G_x}{\partial x}\right)+\left(\frac{\partial F_y}{\partial y}+\frac{\partial G_y}{\partial y}\right)+\left(\frac{\partial F_z}{\partial z}+\frac{\partial G_z}{\partial z}\right)\) \(\nabla \cdot(\mathbf{F}+\mathbf{G})=(\nabla \cdot \mathbf{F})+(\nabla \cdot\mathbf{G})\)
02

Property b: Curl of the Sum

To prove this property, we will use the definition of curl: \(\nabla \times(\mathbf{F}+\mathbf{G})=\det\begin{pmatrix}\hat{\imath} & \hat{\jmath} & \hat{k}\\ \frac{\partial}{\partial x} & \frac{\partial}{\partial y} & \frac{\partial}{\partial z}\\ F_x+G_x & F_y+G_y & F_z+G_z\end{pmatrix}\) Expand the determinant using cofactor expansion and apply properties of partial derivatives: \(\nabla \times(\mathbf{F}+\mathbf{G})=(\nabla \times \mathbf{F})+(\nabla \times \mathbf{G})\)
03

Property c: Divergence of a Scaled Vector Field

To prove this property, we will use the definition of divergence: \(\nabla \cdot(c \mathbf{F})=\frac{\partial(cF_x)}{\partial x}+\frac{\partial(cF_y)}{\partial y}+\frac{\partial(cF_z)}{\partial z}\) Now, we will apply the properties of the partial derivative and constants: \(\nabla \cdot(c \mathbf{F})=c\left(\frac{\partial F_x}{\partial x}+\frac{\partial F_y}{\partial y}+\frac{\partial F_z}{\partial z}\right)=c(\nabla \cdot \mathbf{F})\)
04

Property d: Curl of a Scaled Vector Field

To prove this property, we will use the definition of curl: \(\nabla \times(c \mathbf{F})=\det\begin{pmatrix}\hat{\imath} & \hat{\jmath} & \hat{k}\\ \frac{\partial}{\partial x} & \frac{\partial}{\partial y} & \frac{\partial}{\partial z}\\ cF_x & cF_y & cF_z\end{pmatrix}\) Expand the determinant using cofactor expansion and apply properties of partial derivatives and constants: \(\nabla \times(c \mathbf{F})=c(\nabla \times \mathbf{F})\) We have proven all four properties using the definitions of divergence and curl.

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Most popular questions from this chapter

Prove Green's First Identity for twice differentiable scalar-valued functions \(u\) and \(v\) defined on a region \(D\) : $$\iiint_{D}\left(u \nabla^{2} v+\nabla u \cdot \nabla v\right) d V=\iint_{S} u \nabla v \cdot \mathbf{n} d S$$ where \(\nabla^{2} v=\nabla \cdot \nabla v .\) You may apply Gauss' Formula in Exercise 48 to \(\mathbf{F}=\nabla v\) or apply the Divergence Theorem to \(\mathbf{F}=u \nabla v\)

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\(A\) scalar-valued function \(\varphi\) is harmonic on a region \(D\) if \(\nabla^{2} \varphi=\nabla \cdot \nabla \varphi=0\) at all points of \(D\) Show that if \(\varphi\) is harmonic on a region \(D\) enclosed by a surface \(S\) then $$\iint_{S} \nabla \varphi \cdot \mathbf{n} d S=0$$

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Prove Green's Second Identity for scalar-valued functions \(u\) and \(v\) defined on a region \(D\) : $$\iiint_{D}\left(u \nabla^{2} v-v \nabla^{2} u\right) d V=\iint_{S}(u \nabla v-v \nabla u) \cdot \mathbf{n} d S$$ (Hint: Reverse the roles of \(u\) and \(v\) in Green's First Identity.)
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