Question 4.80

\({\bf V}(x,y,z)=e^{xy}{\bf i}-e^{xy}{\bf j}+e^{yz}{\bf k}\)

Question 4.81

\({\bf V} (x,y,z)=yz{\bf i}+xz{\bf j}+xy{\bf k}\)

Question 4.82

\({\bf V}(x,y,z)=x{\bf i}+(y+\cos x){\bf j}+(z+e^{xy}){\bf k}\)

Question 4.83

\({\bf V}(x,y,z)=x^2{\bf i}+(x+y)^2{\bf j}+(x+y+z)^2{\bf k}\)

Question 4.84

Figure 4.41 shows some flow lines and moving regions for a fluid moving in the plane-field velocity field \({\bf V}\). Where is div \({\bf V} > 0\), and also where is div \({\bf V} < 0\)?

Question 4.85

Let \(V(x,y,z)=x{\bf i}\) be the velocity field of a fluid in space. Relate the sign of the divergence with the rate of change of volume under the flow.

Question 4.86

Sketch a few flow lines for \({\bf F}(x,y)=y{\bf i}\). Calculate \(\nabla\,{\cdot}\, {\bf F}\) and explain why your answer is consistent with your sketch.

Question 4.87

Sketch a few flow lines for \({\bf F}(x,y)=-3x{\bf i}-y{\bf j}\). Calculate \(\nabla \,{\cdot}\, {\bf F}\) and explain why your answer is consistent with your sketch.

Question 4.88

\({\bf F}(x,y)=x^3{\bf i}-x\sin\, (xy){\bf j}\)

Question 4.89

\({\bf F}(x,y)=y{\bf i}-x{\bf j}\)

Question 4.90

\({\bf F}(x,y)=\sin\, (xy){\bf i} -\cos\, (x^2y){\bf j}\)

Question 4.91

\({\bf F}(x,y)=xe^y{\bf i}-[\,y/(x+y)]{\bf j}\)

Question 4.92

\({\bf F}(x,y,z)=x{\bf i}+y{\bf j}+z{\bf k}\)

Question 4.93

\({\bf F}(x,y,z)=yz{\bf i}+xz{\bf j}+xy{\bf k}\)

Question 4.94

\({\bf F}(x,y,z)=(x^2+y^2+z^2)(3{\bf i}+4{\bf j}+5{\bf k})\)

Question 4.95

\({\bf F}(x,y,z)=\displaystyle\frac{yz{\bf i}-xz{\bf j}+xy{\bf k}}{x^2+y^2+z^2}\)

Question 4.96

\({\bf F}(x,y)=\sin x{\bf i}+\cos x{\bf j}\)

Question 4.97

\({\bf F}(x,y)=y{\bf i}-x{\bf j}\)

Question 4.98

\({\bf F}(x,y)=xy{\bf i}+(x^2-y^2){\bf j}\)

Question 4.99

\({\bf F}(x,y)=x{\bf i}+y{\bf j}\)

Question 4.100

Let \(\textbf{F}(x,y,z)=(x^2,x^2y,z+zx)\).

• (a) Verify that \(\nabla \cdot (\nabla \times \textbf{F}) =0\).
• (b) Can there exist a function \(f \colon \mathbb R^3 \to \mathbb R\) such that \(\textbf{F}=\nabla f\)? Explain.

Question 4.101

• (a) Which of the vector fields in Exercises 13–16 could be gradient fields?
• (b) Which of the vector fields in Exercises 9–12 could be the curl of some vector field \(\textbf{V} \colon \mathbb R^3 \to \mathbb R^3\)?

Question 4.102

Let \(\textbf{F}(x, y, z)=(e^{xz}, \sin(xy), x^5y^3z^2)\).

• (a) Find the divergence of \(\textbf{F}\).
• (b) Find the curl of \(\textbf{F}\).

Question 4.103

Suppose \(f\colon \mathbb R^3 \to \mathbb R\) is a \(C^2\) scalar function. Which of the following expressions are meaningful, and which are nonsense? For those which are meaningful, decide whether the expression defines a scalar function or a vector field.

• (a) curl(grad\(f\))
• (b) grad(curl \(f\)))
• (c) div(grad \(f\))
• (d) grad(div \(f\))
• (e) curl (div \(f\))
• (f) div(curl \(f\))

Question 4.104

Suppose \(\textbf{F}\colon \mathbb R^3 \to \mathbb R^3\) is a \(C^2\) vector field. Which of the following expressions are meaningful, and which are nonsense? For those which are meaningful, decide whether the expression defines a scalar function or a vector field.

• (a) curl(grad \(\textbf{F}\))
• (b) grad(curl \(\textbf{F}\)))
• (c) div(grad \(\textbf{F}\))
• (d) grad(div \(\textbf{F}\))
• (e) curl (div \(\textbf{F}\))
• (f) div(curl \(\textbf{F}\))

Question 4.105

Suppose \(f, g, h \colon \mathbb R \to \mathbb R\) are differentiable. Show that the vector field \(\textbf{F}(x, y, z)= \big( f(x), g(y), h(z) \big)\) is irrotational.

Question 4.106

Suppose \(f, g, h \colon \mathbb R^2 \to \mathbb R\) are differentiable. Show that the vector field \(\textbf{F}(x, y, z)= \big( f(y, z), g(x, z), h(x, y) \big)\) has zero divergence.

Question 4.107

Prove identity 13 in the list of vector identities.

Question 4.108

\(f(x,y,z)=\sqrt{x^2+y^2+z^2}\)

Question 4.109

\(f(x,y,z)=xy+yz+xz\)

Question 4.110

\(f(x,y,z)=1/(x^2+y^2+z^2)\)

Question 4.111

\(f(x,y,z)=x^2y^2+y^2z^2\)

Question 4.112

Show that \({\bf F}=y(\cos x){\bf i}+x(\sin y){\bf j}\) is not a gradient vector field.

Question 4.113

Show that \({\bf F}=(x^2+y^2){\bf i}-2xy{\bf j}\) is not a gradient field.

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Question 4.114

Prove identity 10 in the list of vector identities.

Question 4.115

Suppose that \(\nabla \,{\cdot}\, {\bf F}= 0\) and \(\nabla \,{\cdot}\, {\bf G}=0\). Which of the following necessarily have zero divergence?

• (a) \({\bf F}+{\bf G}\)
• (b) \({\bf F}\times {\bf G}\)

Question 4.116

Let \({\bf F}=2xz^2{\bf i}+{\bf j}+y^3 zx{\bf k}\) and \(f=x^2y\). Compute the following quantities.

• (a) \(\nabla f\)
• (b) \(\nabla \times {\bf F}\)
• (c) \({\bf F}\times \nabla f\)
• (d) \({\bf F} \,{\cdot}\, (\nabla f)\)

Question 4.117

Let \({\bf r}(x,y,z)=(x,y,z)\) and \(r=\sqrt{x^2+y^2+z^2}=\|{\bf r}\|\). Prove the following identities.

• (a) \(\nabla (1/r)=-{\bf r}/r^3,r\neq 0\); and, in general, \(\nabla (r^n)=nr^{n-2}{\bf r}\) and \(\nabla (\log r)={\bf r}/r^2\).
• (b) \(\nabla^2 (1/r)=0,r\neq 0;\) and, in general, \(\nabla^2 r^n=n(n+1)\,r^{n-2}.\)
• (c) \(\nabla \,{\cdot}\, ({\bf r}/r^3)=0;\) and, in general, \(\nabla \,{\cdot}\, (r^n {\bf r})=(n+3)\,r^n\).
• (d) \(\nabla \times {\bf r} = {\bf 0};\) and, in general, \(\nabla \times (r^n {\bf r})={\bf 0}\).

Question 4.118

Does \(\nabla \times {\bf F}\) have to be perpendicular to \({\bf F}\)?

Question 4.119

Let \({\bf F}(x,y,z)=3 x^2 y{\bf i}+(x^3+y^3){\bf j}\).

• (a) Verify that curl \({\bf F}={\bf 0}\).
• (b) Find a function \(f\) such that \({\bf F}=\nabla f\). (Techniques for constructing \(f\) in general are given in Chapter 8. The one in this problem should be sought by trial and error.)

Question 4.120

Show that the real and imaginary parts of each of the following complex functions form the components of an irrotational and incompressible vector field in the plane; here \(i=\sqrt{-1}.\)

• (a) \((x-iy)^2\)
• (b) \((x-iy)^3\)
• (c) \(e^{x-iy}=e^x(\cos y-i\sin y)\)