?Determine whether or not the vector field is conservative. If it is conservative, find

Find (a) the curl and (b) the divergence of the vector field.

Find (a) the curl and (b) the divergence of the vector field.

Find (a) the curl and (b) the divergence of the vector field. \(\mathbf{F}(x,\ y,\ z)=xye^{z\ }\mathbf{i}+yze^x\mathbf{\ k}\)

Find (a) the curl and (b) the divergence of the vector field. \(F(x, y, z)=\sin y z i+\sin z x j+\sin x y k\) Equation Transcription: i + j + k Text Transcription: F(x, y, z) = sin yz i + sin zx j + sin xy k

Find (a) the curl and (b) the divergence of the vector field.

Find (a) the curl and (b) the divergence of the vector field. \(\mathbf{F}(x, y, z)=\ln (2 y+3 z) \mathbf{i}+\ln (x+3 z) \mathbf{j}+\ln (x+2 y) \mathbf{k}\)

Find (a) the curl and (b) the divergence of the vector field. \(F(x, y, z)=\left\langle e^{x} \sin y, e^{y} \sin z, e^{z} \sin x\right\rangle\)

Find (a) the curl and (b) the divergence of the vector field. \(\mathbf{F}(x, y, z)=\langle\arctan (x y), \arctan (y z), \arctan (z x)\rangle\)

The vector field \(F\) is shown in the \(xy\)-plane and looks the same in all other horizontal planes. (In other words, \(F\) is independent of \(z\) and its \(z\)-component is 0. ) (a) Is div \(F\) positive, negative, or zero at \(P\) ? Explain. (b) Determine whether curl \(F=0\). If not, in which direction does curl \(F\) point at \(F\) ? Equation Transcription: Text Transcription: F xy F z z F P F = 0 F P

The vector field \(F\) is shown in the \(xy\)-plane and looks the same in all other horizontal planes. (In other words, \(F\) is independent of \(z\) and its \(z\)-component is 0. ) (a) Is div \(F\) positive, negative, or zero at \(P\) ? Explain. (b) Determine whether curl \(F=0\). If not, in which direction does curl \(F\) point at \(F\) ? Equation Transcription: Text Transcription: F xy F z z F P F = 0 F P

The vector field \(\mathbf{F}\) is shown in the xy-plane and looks the same in all other horizontal planes. (In other words, \(\mathbf{F}\) is independent of z and its z-component is 0 .) (a) Is div \(\mathbf{F}\) positive, negative, or zero at P? Explain. (b) Determine whether curl \(\mathbf{F}=\mathbf{0}\). If not, in which direction does \(\operatorname{curl} \mathbf{F}\) point at P?

The vector field \(F\) is shown in the \(xy\)-plane and looks the same in all other horizontal planes. (In other words, \(F\) is independent of \(z\) and its \(z\)-component is 0. ) (a) Is div \(F\) positive, negative, or zero at \(P\) ? Explain. (b) Determine whether curl \(F=0\). If not, in which direction does curl \(F\) point at \(F\) ? Equation Transcription: Text Transcription: F xy F z z F P F = 0 F P

(a) Verify Formula 3 for \(f(x,y,z)=\sin xy\ z\). (b) Verify Formula 11 for \(F(x, y, z)=x y z^{2} i+x^{2} y z^{3} j+y^{2} k\) Equation Transcription: i + j + k Text Transcription: f(x, y, z) = sin xy z F(x, y, z) = xyz^2 i + x^2yz^3 j + y^2 k

Let f be a scalar field and \(\mathbf{F}\) a vector field. State whether each expression is meaningful. If not, explain why. If so, state whether it is a scalar field or a vector field. (a) \(\operatorname{curl} f\) (b) \(\operatorname{grad} f\) (c) \(\operatorname{div} \mathbf{F}\) (d) \(\operatorname{curl}(\operatorname{grad} f)\) (e) \(\operatorname{grad} \mathbf{F}\) (f) \(\operatorname{grad}(\operatorname{div} \mathbf{F})\) (g) \(\operatorname{div}(\operatorname{grad} f)\) (h) \(\operatorname{grad}(\operatorname{div} f)\) (i) \(\operatorname{curl}(\operatorname{curl} \mathbf{F})\) (j) \(\operatorname{div}(\operatorname{div} \mathbf{F})\) (k) \((\operatorname{grad} f) \times(\operatorname{div} \mathbf{F})\) (1) \(\operatorname{div}(\operatorname{curl}(\operatorname{grad} f))\)

Determine whether or not the vector field is conservative. If it is conservative, find a function \(f\) such that \(\mathbf{F}=\nabla f\) \(\mathbf{F}(x, y, z)=\left\langle 2 x y^{3} z^{2}, 3 x^{2} y^{2} z^{2}, 2 x^{2} y^{3} z\right\rangle\)

Determine whether or not the vector field is conservative. If it is conservative, find a function such that .

Determine whether or not the vector field is conservative. If it is conservative, find a function \(f\) such that \(F=\nabla f\) \(F(x, y, z)=\langle\ln y,(x / y)+\ln z, y / z\rangle\)

Determine whether or not the vector field is conservative. If it is conservative, find a function \(f\) such that \(F=\nabla f\) \(F(x, y, z)=y z \sin x y \mathbf{i}+x z \sin x y \mathbf{j}-\cos x y \mathbf{k}\)

Determine whether or not the vector field is conservative. If it is conservative, find a function such that .

Determine whether or not the vector field is conservative. If it is conservative, find a function \(f\) such that \(F=\nabla f\) . \(F(x, y, z)=e^{z} \cos x i+e^{y} \cos z j+\left(e^{z} \sin x-e^{y} \sin z\right) k\) Equation Transcription: = ? i + j + k Text Transcription: F F = nabla f F(x, y, z) = e^z cos x i + e^y cos z j + (e^z sin x - e^y sin z) k

Is there a vector field \(G\) on \(\mathbb{R}^{3}\) such that curl \(G=\langle x \sin y, \cos y, z-x y\rangle\)? Explain. Equation Transcription: ?3 ?? Text Transcription: G mathbbR^3 G = left angle x sin y, cos y, z - xy right angle

Is there a vector field on such that curl ? Explain.

Show that any vector field of the form \(\mathbf{F}(x,\ y,\ z)=f(x)\mathbf{\ i}+g(y)\mathbf{\ j}+h(z)\ \mathbf{k}\) where f, g, h are differentiable functions, is irrotational.

Show that any vector field of the form is incompressible.

Prove the identity, assuming that the appropriate partial derivatives exist and are continuous. If \(f\) is a scalar field and \(F\), \(G\) are vector fields, then \(f F, F \cdot G\), and \(F \times G\) are defined by \((f F)(x, y, z)=f(x, y, z) F(x, y, z)\) \((F \cdot G)(x, y, z)=F(x, y, z) \cdot G(x, y, z)\) \((F \times G)(x, y, z)=F(x, y, z) \times G(x, y, z)\) \(\operatorname{div} F+G)=\operatorname{div} F+\operatorname{div} G\) Equation Transcription: div= div div Text Transcription: f F G fF, F . G F x G (fF)(x, y, z) = f(x, y, z) F (x, y, z ) (F . G)(x, y, z) = F(x, y, z) . G (x, y, z ) (F x G)(x, y, z) = F(x, y, z) x G (x, y, z ) div(F + G) = div F + div G

Prove the identity, assuming that the appropriate partial derivatives exist and are continuous. If \(f\) is a scalar field and \(F\), \(G\) are vector fields, then \(f F, F \cdot G\), and \(F \times G\) are defined by \((f F)(x, y, z)=f(x, y, z) F(x, y, z)\) \((F \cdot G)(x, y, z)=F(x, y, z) \cdot G(x, y, z)\) \((F \times G)(x, y, z)=F(x, y, z) \times G(x, y, z)\) \(\operatorname{curl} F+G)=\operatorname{curl} F+\operatorname{curl} G\) Equation Transcription: curl= curl div Text Transcription: f F G fF, F . G F x G (fF)(x, y, z) = f(x, y, z) F (x, y, z ) (F . G)(x, y, z) = F(x, y, z) . G (x, y, z ) (F x G)(x, y, z) = F(x, y, z) x G (x, y, z ) curl(F + G) = curl F + curl G

Prove the identity, assuming that the appropriate partial derivatives exist and are continuous. If \(f\) is a scalar field and \(F\), \(G\) are vector fields, then \(f F, F \cdot G\), and \(F \times G\) are defined by \((f F)(x, y, z)=f(x, y, z) F(x, y, z)\) \((F \cdot G)(x, y, z)=F(x, y, z) \cdot G(x, y, z)\) \((F \times G)(x, y, z)=F(x, y, z) \times G(x, y, z)\) \(\operatorname{div}(f F)=f d \dot{v} F+F \cdot \nabla f\) Equation Transcription: div= div . ? Text Transcription: f F G fF, F . G F x G (fF)(x, y, z) = f(x, y, z) F (x, y, z ) (F . G)(x, y, z) = F(x, y, z) . G (x, y, z ) (F x G)(x, y, z) = F(x, y, z) x G (x, y, z ) div(fF) = f div F + F . nabla f

Prove the identity, assuming that the appropriate partial derivatives exist and are continuous. If \(f\) is a scalar field and \(F\), \(G\) are vector fields, then \(f F, F \cdot G\), and \(F \times G\) are defined by \((f F)(x, y, z)=f(x, y, z) F(x, y, z)\) \((F \cdot G)(x, y, z)=F(x, y, z) \cdot G(x, y, z)\) \((F \times G)(x, y, z)=F(x, y, z) \times G(x, y, z)\) \(c u r l(f F)=f \operatorname{curl} F+(\nabla f) \times F\) Equation Transcription: curl= curl (?) Text Transcription: f F G fF, F . G F x G (fF)(x, y, z) = f(x, y, z) F (x, y, z ) (F . G)(x, y, z) = F(x, y, z) . G (x, y, z ) (F x G)(x, y, z) = F(x, y, z) x G (x, y, z ) curl(fF)= f curl F + (nabla f) x F

Prove the identity, assuming that the appropriate partial derivatives exist and are continuous. If \(f\) is a scalar field and \(F\), \(G\) are vector fields, then \(f F, F \cdot G\), and \(F \times G\) are defined by \((f F)(x, y, z)=f(x, y, z) F(x, y, z)\) \((F \cdot G)(x, y, z)=F(x, y, z) \cdot G(x, y, z)\) \((F \times G)(x, y, z)=F(x, y, z) \times G(x, y, z)\) \(\operatorname{div}(F \times G)=G \cdot \operatorname{curl} F-F \cdot \operatorname{curl} G\) Equation Transcription: div= . curl . curl Text Transcription: f F G fF, F . G F x G (fF)(x, y, z) = f(x, y, z) F (x, y, z ) (F . G)(x, y, z) = F(x, y, z) . G (x, y, z ) (F x G)(x, y, z) = F(x, y, z) x G (x, y, z ) div(F x G) = G . curl F - F . curl G

Prove the identity, assuming that the appropriate partial derivatives exist and are continuous. If \(f\) is a scalar field and \(F\), \(G\) are vector fields, then \(f F, F \cdot G\), and \(F \times G\) are defined by \((f F)(x, y, z)=f(x, y, z) F(x, y, z)\) \((F \cdot G)(x, y, z)=F(x, y, z) \cdot G(x, y, z)\) \((F \times G)(x, y, z)=F(x, y, z) \times G(x, y, z)\) \(\operatorname{div}(\nabla f \times \nabla g)=0\) Equation Transcription: div(??) = 0 Text Transcription: f F G fF, F . G F x G (fF)(x, y, z) = f(x, y, z) F (x, y, z ) (F . G)(x, y, z) = F(x, y, z) . G (x, y, z ) (F x G)(x, y, z) = F(x, y, z) x G (x, y, z ) div(nabla f x nabla g) = 0

Prove the identity, assuming that the appropriate partial derivatives exist and are continuous. If f is a scalar field and \(\mathbf{F}\), \(\mathbf{G}\) are vector fields, then \(f \mathbf{F}, \mathbf{F} \cdot \mathbf{G}\), and \(\mathbf{F} \times \mathbf{G}\) are defined by \(\begin{aligned}(f \mathbf{F})(x, y, z) &=f(x, y, z) \mathbf{F}(x, y, z) \\(\mathbf{F} \cdot\mathbf{G})(x, y, z) &=\mathbf{F}(x, y, z) \cdot \mathbf{G}(x, y, z) \\(\mathbf{F} \times\mathbf{G})(x, y, z) &=\mathbf{F}(x, y, z) \times \mathbf{G}(x, y, z)\end{aligned}\) \(\operatorname{curl}(\operatorname{curl} \mathbf{F})=\operatorname{grad}(\operatorname{div} \mathbf{F})-\nabla^2 \mathbf{F}\)

Let and . Verify each identity. (a) (b) (c)

Let \(r=x i+y j+z k\) and \(r=|r|\). Verify each identity. (a) \(\nabla r=r / r\) (b) \(\nabla \times r=0\) (c) \(\nabla(1 / r)=-r / r^{3}\) (d) \(\nabla l n r=r / r^{2}\) Equation Transcription: ? ? ? ? Text Transcription: r=xi+yj+zk r=|r| nable r=r/r nabla x r=0 nabla(1/r)=-r/r^3 nabla lnr=r/r^2

Let \(r=x i+y j+z k\) and \(r=|r|\). If \(F=r / r^{p}\), find div \(F\). Is there a value of \(p\) for which \(d i v F=0\)? Equation Transcription: Text Transcription: r=xi+yj+zk r=|r| F=r/r^p F p divF=0

Use Green's Theorem in the form of Equation 13 to prove Green's first identity: \(\iint_{D} f \nabla^{2} g d A=\oint_{c} f(\nabla g) \cdot n d s-\iint_{D} \nabla f \cdot \nabla g d A\) where D and C satisfy the hypotheses of Green's Theorem and the appropriate partial derivatives of f and g exist and are continuous. (The quantity \(\nabla g \cdot n=D_{n} g\) occurs in the line integral; it is the directional derivative in the direction of the normal vector n and is called the normal derivative of g.) Equation Transcription: ? ? (?g)??? ? Text Transcription: Double Integral_D f nabla^2 gdA = contour integral_C f(nabla g) times nds - Double Integral_D nabla f times nabla gdA nabla g times n = D_n g

Use Green's first identity (Exercise 35 ) to prove Green's second identity: \(\iint_{D}\left(f \nabla^{2} g-g \nabla^{2} f\right) d A=\oint_{C}(f \nabla g-g \nabla f) \cdot n d s\) where D and C satisfy the hypotheses of Green's Theorem and the appropriate partial derivatives of f and g exist and are continuous. Equation Transcription: ? ?? (f?g-g?f) Text Transcription: Double Intral_D (f nabla^2 g-g nabla^2 f) dA = contour integral_C (f nabla g-g nabla f) times nds

Recall from Section that a function is called harmonic on if it satisfies Laplace's equation, that is, on . Use Green's first identity (with the same hypotheses as in Exercise 35 ) to show that if is harmonic on , then Here is the normal derivative of defined in Exercise

Use Green's first identity to show that if f is harmonic on D, and if \(f(x, \ y)=0\) on the boundary curve C, then \(\iint_{D}|\nabla f|^{2} \ d A=0\). (Assume the same hypotheses as in Exercise 35.)

This exercise demonstrates a connection between the curl vector and rotations. Let B be a rigid body rotating about the z-axis. The rotation can be described by the vector \(\mathbf{w}=\omega \mathbf{k}\), where \(\omega\) is the angular speed of B, that is, the tangential speed of any point P in B divided by the distance d from the axis of rotation. Let \(\mathbf{r}=\langle x, y, z\rangle\) be the position vector of P. (a) By considering the angle \(\theta\) in the figure, show that the velocity field of B is given by \(\mathbf{v}=\mathbf{w} \times \mathbf{r}\). (b) Show that \(\mathbf{v}=-\omega y \mathbf{i}+\omega x \mathbf{j}\). (c) Show that \(\operatorname{curl} \mathbf{v}=2 \mathbf{w}\).

Maxwell's equations relating the electric field E and magnetic field H as they vary with time in a region containing no charge and no current can be stated as follows: div E = 0 div H = 0 \(\text {curl} \mathbf{\ E}=-\frac{1}{c}\frac{\partial\mathbf{H}}{\partial t}\) \(\text {curl} \mathbf{\ H}=\frac{1}{c}\frac{\partial\mathbf{E}}{\partial t}\) where c is the speed of light. Use these equations to prove the following: (a) \(\nabla \times(\nabla \times \mathbf{E})=-\frac{1}{c^{2}} \frac{\partial^{2} \mathbf{E}}{\partial t^{2}}\) (b) \(\nabla \times(\nabla \times \mathbf{H})=-\frac{1}{c^{2}} \frac{\partial^{2} \mathbf{H}}{\partial t^{2}}\) (c) \(\nabla^{2} \mathbf{E}=\frac{1}{c^{2}} \frac{\partial^{2} \mathbf{E}}{\partial t^{2}}\) [Hint: Use Exercise 31.] (d) \(\nabla^{2} \mathbf{H}=\frac{1}{c^{2}} \frac{\partial^{2} \mathbf{H}}{\partial t^{2}}\)

We have seen that all vector fields of the form \(F=\nabla g\) satisfy the equation curl \(F=0\) and that all vector fields of the form \(F=\operatorname{curl} G\) satisfy the equation \(\operatorname{div} F=0\) (assuming continuity of the appropriate partial derivatives). This suggests the question: are there any equations that all functions of the form \(f=\operatorname{div} G\) must satisfy? Show that the answer to this question is "no" by proving that \(e\) very continuous function \(f\) on \(R^{3}\) is the divergence of some vector field. [Hint: Let \(G(x, y, z)=\langle g(x, y, z), 0,0\rangle\), where \(g(x, y, z)=\int_{0}^{x} f(t, y, z) d t\).] Equation Transcription: ? ?? Text Transcription: F=nabla g F=0 F=curl G divF=0 f=div G e f R^3 G(x,y,z)=langle g(x,y,z),0,0 rangle g(x,y,z)=integral _0 ^x f(t,y,z)dt