Question 14.199

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

Question 14.200

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

Question 14.201

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

Question 14.202

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

Question 14.203

Figure # 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 14.204

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 14.205

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 14.206

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 14.207

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

Question 14.208

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

Question 14.209

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

Question 14.210

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

Question 14.211

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

Question 14.212

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

Question 14.213

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

Question 14.214

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

Question 14.215

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

Question 14.216

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

Question 14.217

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

Question 14.218

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

Question 14.219

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 14.220

• (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 14.221

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 14.222

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 14.223

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 14.224

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 14.225

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 14.226

Prove identity 13 in the list of vector identities.

Question 14.227

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

Question 14.228

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

Question 14.229

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

Question 14.230

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

Question 14.231

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

Question 14.232

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

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Question 14.233

Prove identity 10 in the list of vector identities.

Question 14.234

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 14.235

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 14.236

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 14.237

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

Question 14.238

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 14.239

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) \(e^{x-iy}=e^x(\cos y-i\sin y)\)