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mathematics
calculus 4th

Questions and Answers of Calculus 4th

A hose feeds into a small screen box of volume \(10 \mathrm{~cm}^{3}\) that is suspended in a swimming pool. Water flows across the surface of the box at a rate of \(12 \mathrm{~cm}^{3} /
The electric field due to a unit electric dipole oriented in the \(\mathbf{k}\)-direction is \(\mathbf{E}=abla\left(z / r^{3}ight)\), where \(r=\) \(\left(x^{2}+y^{2}+z^{2}ight)^{1 / 2}\) (Figure
Let \(\mathbf{E}\) be the electric field due to a long, uniformly charged rod of radius \(R\) with charge density \(\delta\) per unit length (Figure 21). By symmetry, we may assume that
Let \(\mathcal{W}\) be the region between the sphere of radius 4 and the cube of side 1, both centered at the origin. What is the flux through the boundary \(\mathcal{S}=\partial \mathcal{W}\) of a
Let \(\mathcal{W}\) be the region between the sphere of radius 3 and the sphere of radius 2 , both centered at the origin. Use the Divergence Theorem to calculate the flux of \(\mathbf{F}=x
Let \(f\) be a scalar function and \(\mathbf{F}\) be a vector field. Prove the following Product Rule for Divergence:\[\operatorname{div}(f \mathbf{F})=f \operatorname{div}(\mathbf{F})+abla f \cdot
Let \(\mathbf{F}\) and \(\mathbf{G}\) be vector fields. Prove the the following Product Rule for Divergence:\[\operatorname{div}(\mathbf{F} \times \mathbf{G})=\operatorname{curl}(\mathbf{F}) \cdot
A vector field \(\mathbf{F}\) is incompressible if \(\operatorname{div}(\mathbf{F})=0\) and is irrotational if \(\operatorname{curl}(\mathbf{F})=\mathbf{0}\).Let \(\mathbf{F}\) be an incompressible
A vector field \(\mathbf{F}\) is incompressible if \(\operatorname{div}(\mathbf{F})=0\) and is irrotational if \(\operatorname{curl}(\mathbf{F})=\mathbf{0}\).Prove that the cross product of two
\(\Delta\) denotes the Laplace operator defined byProve the identity = + dx2 +
\(\Delta\) denotes the Laplace operator defined byProve the identity\[\operatorname{curl}(\operatorname{curl}(\mathbf{F}))=abla(\operatorname{div}(\mathbf{F}))-\Delta \mathbf{F}\]where \(\Delta
\(\Delta\) denotes the Laplace operator defined byA function \(\varphi\) satisfying \(\Delta \varphi=0\) is called harmonic.(a) Show that \(\Delta \varphi=\operatorname{div}(abla \varphi)\) for any
\(\Delta\) denotes the Laplace operator defined byLet \(\mathbf{F}=r^{n} \mathbf{e}_{r}\), where \(n\) is any number, \(r=\left(x^{2}+y^{2}+z^{2}ight)^{1 / 2}\), and \(\mathbf{e}_{r}=r^{-1}\langle x,
Let \(\mathcal{S}\) be the boundary surface of a region \(\mathcal{W}\) in \(\mathbf{R}^{3}\), and let \(D_{\mathbf{n}} \varphi\) denote the directional derivative of \(\varphi\), where
Assume that \(\varphi\) is harmonic. Show that \(\operatorname{div}(\varphi abla \varphi)=\|abla \varphi\|^{2}\) and conclude that\[\iint_{\mathcal{S}} \varphi D_{\mathbf{n}} \varphi d
Let \(\mathbf{F}=\langle P, Q, Rangle\) be a vector field defined on \(\mathbf{R}^{3}\) such that \(\operatorname{div}(\mathbf{F})=0\). Use the following steps to show that \(\mathbf{F}\) has a
Show that\[\mathbf{F}(x, y, z)=\left\langle 2 y-1,3 z^{2}, 2 x yightangle\]has a vector potential and find one.
Show that\[\mathbf{F}(x, y, z)=\left\langle 2 y e^{z}-x y, y, y z-zightangle\]has a vector potential and find one.
Let \(\mathbf{F}(x, y)=\left\langle x+y^{2}, x^{2}-yightangle\), and let \(C\) be the unit circle, oriented counterclockwise. Evaluate \(\oint_{C} \mathbf{F} \cdot d \mathbf{r}\) directly as a line
Let \(\partial \mathcal{R}\) be the boundary of the rectangle in Figure 1, and let \(\partial \mathcal{R}_{1}\) and \(\partial \mathcal{R}_{2}\) be the boundaries of the two triangles, all oriented
Calculate the flux of the vector field \(\mathbf{E}(x, y, z)=\langle 0,0, xangle\) through the part of the ellipsoid\[4 x^{2}+9 y^{2}+z^{2}=36\]where \(z \geq 3, x \geq 0, y \geq 0\). Hint: Use the
Which vector field \(\mathbf{F}\) is being integrated in the line integral \(\oint x^{2} d y-e^{y} d x\) ?
Indicate which of the following vector fields possess this property: For every simple closed curve \(C\),\(\oint_{C} \mathbf{F} \cdot d \mathbf{r}\) is equal to the area enclosed by \(C\).(a)
Let \(A\) be the area enclosed by a simple closed curve \(C\), and assume that \(C\) is oriented counterclockwise. Indicate whether the value of each integral is \(0,-A\), or \(A\).(a) \(\oint_{C} x
Verify Green's Theorem for the line integral \(\oint_{C} x y d x+y d y\), where \(C\) is the unit circle, oriented counterclockwise. THEOREM 1 Green's Theorem Let D be a domain whose boundary ID is a
Let \(I=\oint_{C} \mathbf{F} \cdot d \mathbf{r}\), where \(\mathbf{F}(x, y)=\left\langle y+\sin x^{2}, x^{2}+e^{y^{2}}ightangle\) and \(C\) is the circle of radius 4 centered at the origin.(a) Which
Use Green's Theorem to evaluate the line integral. Orient the curve counterclockwise unless otherwse indicated.\(\oint_{C} y^{2} d x+x^{2} d y\), where \(C\) is the boundary of the square that is
Use Green's Theorem to evaluate the line integral. Orient the curve counterclockwise unless otherwse indicated.\(\oint_{C} y^{2} d x+x^{2} d y\), where \(C\) is the boundary of the square \(-1 \leq x
Use Green's Theorem to evaluate the line integral. Orient the curve counterclockwise unless otherwse indicated.\(\oint_{C} 5 y d x+2 x d y\), where \(C\) is the triangle with vertices
Use Green's Theorem to evaluate the line integral. Orient the curve counterclockwise unless otherwse indicated.\(\oint_{C} e^{2 x+y} d x+e^{-y} d y\), where \(C\) is the triangle with vertices
Use Green's Theorem to evaluate the line integral. Orient the curve counterclockwise unless otherwse indicated.\(\oint_{C} x^{2} y d x\), where \(C\) is the unit circle centered at the origin THEOREM
Use Green's Theorem to evaluate the line integral. Orient the curve counterclockwise unless otherwse indicated.\(\oint_{C} \mathbf{F} \cdot d \mathbf{r}\), where \(\mathbf{F}(x, y)=\left\langle x+y,
Use Green's Theorem to evaluate the line integral. Orient the curve counterclockwise unless otherwse indicated.\(\oint_{C} \mathbf{F} \cdot d \mathbf{r}\), where \(\mathbf{F}(x, y)=\left\langle
Use Green's Theorem to evaluate the line integral. Orient the curve counterclockwise unless otherwse indicated.\(\oint_{C}(\ln x+y) d x-x^{2} d y\), where \(C\) is the rectangle with vertices
Use Green's Theorem to evaluate the line integral. Orient the curve counterclockwise unless otherwse indicated.The line integral of \(\mathbf{F}(x, y)=\left\langle e^{x+y}, e^{x-y}ightangle\) along
Use Green's Theorem to evaluate the line integral. Orient the curve counterclockwise unless otherwse indicated.∫Cxydx+(x2+x)dy∫Cxydx+(x2+x)dy, where CC is the path in Figure 17 THEOREM 1 Green's
Use Green's Theorem to evaluate the line integral. Orient the curve counterclockwise unless otherwse indicated.Let \(\mathbf{F}(x, y)=\left\langle 2 x e^{y}, x+x^{2} e^{y}ightangle\) and let \(C\) be
Compute the line integral of \(\mathbf{F}(x, y)=\left\langle x^{3}, 4 xightangle\) along the path from \(A\) to \(B\) in Figure 19. To save work, use Green's Theorem to relate this line integral to
Evaluate \(I=\int_{C}(\sin x+y) d x+(3 x+y) d y\) for the nonclosed path \(A B C D\) in Figure 20. Use the method of Exercise 14.Data From Exercise 14Compute the line integral of \(\mathbf{F}(x,
Use \(\oint_{C} y d x\) to compute the area of the ellipse \(\left(\frac{x}{a}ight)^{2}+\left(\frac{y}{b}ight)^{2}=1\).
Use \(\frac{1}{2} \oint_{C} x d y-y d x\) to compute the area of the ellipse \(\left(\frac{x}{a}ight)^{2}+\left(\frac{y}{b}ight)^{2}=1\).
Use one of the formulas in Eq. (6) to calculate the area of the given region.The circle of radius 3 centered at the origin = fxdy = f-ydx = 1/ fxdy - ydx 2 area enclosed by C =
Use one of the formulas in Eq. (6) to calculate the area of the given region.The triangle with vertices \((0,0),(1,0)\), and \((1,1)\) = fxdy = fydx = 1 fxdy - ydx 2 area enclosed by C =
Use one of the formulas in Eq. (6) to calculate the area of the given region.The region between the \(x\)-axis and the cycloid parametrized by \(\mathbf{r}(t)=\langle t-\sin t, 1-\cos tangle\) for
Use one of the formulas in Eq. (6) to calculate the area of the given region.The region between the graph of \(y=x^{2}\) and the \(x\)-axis for \(0 \leq x \leq 2\) = fxdy = fydx = 1 fxdy - ydx 2 area
A square with vertices \((1,1),(-1,1),(-1,-1)\), and \((1,-1)\) has area 4 . Calculate this area three times using the formulas in Eq. (6). = fxdy = f -y dx = 1 f xdy - ydx 2 area enclosed by C =
Let \(x^{3}+y^{3}=3 x y\) be the folium of Descartes (Figure 22).(a) Show that the folium has a parametrization in terms of \(t=y / x\) given by\[x=\frac{3 t}{1+t^{3}}, \quad y=\frac{3
Find a parametrization of the lemniscate \(\left(x^{2}+y^{2}ight)^{2}=x y\) (see Figure 23) by using \(t=y / x\) as a parameter (see Exercise 23). Then use Eq. (6) to find the area of one loop of the
The Centroid via Boundary Measurements The centroid (see Section 15.5) of a domain \(\mathcal{D}\) enclosed by a simple closed curve \(C\) is the point with coordinates \((\bar{x},
Use the result of Exercise 25 to compute the moments of the semicircle x2+y2=R2,y≥0x2+y2=R2,y≥0 as line integrals. Verify that the centroid is (0,4R/(3π))(0,4R/(3π)).Data From Exercise 25The
Let \(C_{R}\) be the circle of radius \(R\) centered at the origin. Use the general form of Green's Theorem to determine \(\oint_{C_{2}} \mathbf{F} \cdot d \mathbf{r}\), where \(\mathbf{F}\) is a
Referring to Figure 24 , suppose that \(\oint_{C_{2}} \mathbf{F} \cdot d \mathbf{r}=12\). Use Green's Theorem to determine \(\oint_{C_{1}} \mathbf{F} \cdot d \mathbf{r}\), assuming that
Referring to Figure 25, suppose that\[\oint_{C_{2}} \mathbf{F} \cdot d \mathbf{r}=3 \pi, \quad \oint_{C_{3}} \mathbf{F} \cdot d \mathbf{r}=4 \pi\]Use Green's Theorem to determine the circulation of
Let \(\mathbf{F}\) be the vector field\[\mathbf{F}(x, y)=\left\langle\frac{x}{x^{2}+y^{2}}, \frac{y}{x^{2}+y^{2}}ightangle\]and assume that \(C_{R}\) is the circle of radius \(R\) centered at the
We refer to the integrand that occurs in Green's Theorem and that appears as\[\operatorname{curl}_{z}(\mathbf{F})=\frac{\partial F_{2}}{\partial x}-\frac{\partial F_{1}}{\partial y}\]For the vector
We refer to the integrand that occurs in Green's Theorem and that appears as\[\operatorname{curl}_{z}(\mathbf{F})=\frac{\partial F_{2}}{\partial x}-\frac{\partial F_{1}}{\partial y}\]Estimate the
We refer to the integrand that occurs in Green's Theorem and that appears as\[\operatorname{curl}_{z}(\mathbf{F})=\frac{\partial F_{2}}{\partial x}-\frac{\partial F_{1}}{\partial y}\]Estimate
We refer to the integrand that occurs in Green's Theorem and that appears ascurlz(F)=∂F2∂x−∂F1∂ycurlz⁡(F)=∂F2∂x−∂F1∂yLet FF be a velocity field. Estimate the circulation of FF
Let \(C_{R}\) be the circle of radius \(R\) centered at the origin. Use Green's Theorem to find the value of \(R\) that maximizes \(\oint_{C_{R}} y^{3} d x+x d y\).
Calculate \(\operatorname{div}(\mathbf{F})\) and \(\operatorname{curl}(\mathbf{F})\).\(\mathbf{F}=\left\langle e^{x+y}, e^{y+z}, x y zightangle\)
Calculate \(\operatorname{div}(\mathbf{F})\) and \(\operatorname{curl}(\mathbf{F})\).\(\mathbf{F}=abla\left(e^{-x^{2}-y^{2}-z^{2}}ight)\)
Calculate \(\operatorname{div}(\mathbf{F})\) and \(\operatorname{curl}(\mathbf{F})\).\(\mathbf{e}_{r}=r^{-1}\langle x, y, zangle\left(r=\sqrt{x^{2}+y^{2}+z^{2}}ight)\)
Show that if \(F_{1}, F_{2}\), and \(F_{3}\) are differentiable functions of one variable, then\[\operatorname{curl}\left(\left\langle F_{1}(x), F_{2}(y), F_{3}(z)ightangleight)=\mathbf{0}\]Use this
Give an example of a nonzero vector field \(\mathbf{F}\) such that \(\operatorname{curl}(\mathbf{F})=\mathbf{0}\) and \(\operatorname{div}(\mathbf{F})=0\).
Verify the identity \(\operatorname{div}(\operatorname{curl} \mathbf{F})=0\) for the vector fields \(\mathbf{F}=\left\langle x z, y e^{x}, y zightangle\) and \(\mathbf{G}=\left\langle z^{2}, x y^{3},
Determine whether the vector field is conservative, and if so, find a potential function.\(\mathbf{F}(x, y)=\left\langle x^{2} y, y^{2} xightangle\)
Determine whether the vector field is conservative, and if so, find a potential function.\(\mathbf{F}(x, y)=\left\langle 4 x^{3} y^{5}, 5 x^{4} y^{4}ightangle\)
Determine whether the vector field is conservative, and if so, find a potential function.\(\mathbf{F}(x, y, z)=\left\langle\sin x, e^{y}, zightangle\)
Determine whether the vector field is conservative, and if so, find a potential function.\(\mathbf{F}(x, y, z)=\left\langle 2,4, e^{z}ightangle\)
Determine whether the vector field is conservative, and if so, find a potential function.\(\mathbf{F}(x, y, z)=\left\langle x y z, \frac{1}{2} x^{2} z, 2 z^{2} yightangle\)
Determine whether the vector field is conservative, and if so, find a potential function.\(\mathbf{F}(x, y)=\left\langle y^{4} x^{3}, x^{4} y^{3}ightangle\)
Determine whether the vector field is conservative, and if so, find a potential function.\(\mathbf{F}(x, y, z)=\left\langle\frac{y}{1+x^{2}}, \tan ^{-1} x, 2 zightangle\)
Determine whether the vector field is conservative, and if so, find a potential function.\(\mathbf{F}(x, y, z)=\left\langle\frac{2 x y}{x^{2}+z}, \ln \left(x^{2}+zight), \frac{y}{x^{2}+z}ightangle\)
Determine whether the vector field is conservative, and if so, find a potential function.\(\mathbf{F}(x, y, z)=\left\langle x e^{2 x}, y e^{2 z}, z e^{2 y}ightangle\)
Find a conservative vector field of the form \(\mathbf{F}=\langle g(y), h(x)angle\) such that \(\mathbf{F}(0,0)=\langle 1,1angle\), where \(g(y)\) and \(h(x)\) are differentiable functions. Determine
Compute the line integral \(\int_{C} f(x, y) d s\) for the given function and path or curve.\(f(x, y)=x y\), the path \(\mathbf{r}(t)=\langle t, 2 t-1angle\) for \(0 \leq t \leq 1\)
Compute the line integral \(\int_{C} f(x, y) d s\) for the given function and path or curve.\(f(x, y)=x-y\), the unit semicircle \(x^{2}+y^{2}=1, y \geq 0\)
Compute the line integral \(\int_{C} f(x, y) d s\) for the given function and path or curve.\(f(x, y, z)=e^{x}-\frac{y}{2 \sqrt{2} z}, \quad\) the path \(\mathbf{r}(t)=\left\langle\ln t, \sqrt{2} t,
Compute the line integral \(\int_{C} f(x, y) d s\) for the given function and path or curve.\(f(x, y, z)=x+2 y+z\), the helix \(\mathbf{r}(t)=\langle\cos t, \sin t, tangle\) for \(0 \leq t \leq \pi /
Find the total mass of an \(\mathrm{L}\)-shaped rod consisting of the segments \((2 t, 2)\) and \((2,2-2 t)\) for \(0 \leq t \leq 1\) (length in centimeters) with mass density \(\delta(x, y)=x^{2} y
Calculate \(\mathbf{F}=abla f\), where \(f(x, y, z)=x y e^{z}\), and compute \(\int_{C} \mathbf{F} \cdot d \mathbf{r}\), where:(a) \(C\) is any curve from \((1,1,0)\) to \((3, e,-1)\).(b) \(C\) is
Calculate \(\int_{C_{1}} y d x+x^{2} y d y\), where \(C_{1}\) is the oriented curve in Figure 1(A). 3 C (A) 3 -X y 3 (B) C 3 -X
Let F(x,y)=⟨9y−y3,e√y(x2−3x)⟩F(x,y)=⟨9y−y3,ey(x2−3x)⟩, and let C2C2 be the oriented curve in Figure 1(B).(a) Show that FF is not conservative.(b) Show that
Compute the line integral \(\int_{\mathbf{c}} \mathbf{F} \cdot d \mathbf{r}\) for the given vector field and path.\(\mathbf{F}(x, y)=\left\langle\frac{2 y}{x^{2}+4 y^{2}}, \frac{x}{x^{2}+4
Compute the line integral \(\int_{\mathbf{c}} \mathbf{F} \cdot d \mathbf{r}\) for the given vector field and path.\(\mathbf{F}(x, y)=\left\langle 2 x y, x^{2}+y^{2}ightangle\), the part of the unit
Compute the line integral \(\int_{\mathbf{c}} \mathbf{F} \cdot d \mathbf{r}\) for the given vector field and path.\(\mathbf{F}(x, y)=\left\langle x^{2} y, y^{2} z, z^{2} xightangle\), the path
Compute the line integral \(\int_{\mathbf{c}} \mathbf{F} \cdot d \mathbf{r}\) for the given vector field and path.\(\mathbf{F}=abla f\), where \(f(x, y, z)=4 x^{2} \ln \left(1+y^{4}+z^{2}ight),
Consider the line integrals \(\int_{C} \mathbf{F} \cdot d \mathbf{r}\) for the vector fields \(\mathbf{F}\) and paths \(\mathbf{r}\) in Figure 2. Which two of the line integrals appear to have a
Calculate the work required to move an object from \(P=(1,1,1)\) to \(Q=(3,-4,-2)\) against the force field \(\mathbf{F}(x, y, z)=-12 r^{-4}\langle x, y, zangle\) (distance in meters, force in
Find constants \(a, b, c\) such that\[\Phi(u, v)=(u+a v, b u+v, 2 u-c)\]parametrizes the plane \(3 x-4 y+z=5\). Calculate \(\mathbf{T}_{u}, \mathbf{T}_{v}\), and \(\mathbf{N}(u, v)\).
Calculate the integral of \(f(x, y, z)=e^{z}\) over the portion of the plane \(x+2 y+2 z=3\), where \(x, y, z \geq 0\).
Compute \(\iint_{\mathcal{S}} \mathbf{F} \cdot d \mathbf{S}\) for the given oriented surface or parametrized surface.\(\mathbf{F}(x, y, z)=\left\langle z, 0, z^{2}ightangle, \quad \Phi(u, v)=(v \cosh
Let \(\mathcal{S}\) be the surface parametrized by\[\Phi(u, v)=\left(2 u \sin \frac{v}{2}, 2 u \cos \frac{v}{2}, 3 vight)\]for \(0 \leq u \leq 1\) and \(0 \leq v \leq 2 \pi\)(a) Calculate the tangent
Plot the surface with parametrization\[\Phi(u, v)=(u+4 v, 2 u-v, 5 u v)\]for \(-1 \leq v \leq 1,-1 \leq u \leq 1\). Express the surface area as a double integral and use a computer algebra system to
Express the surface area of the surface \(z=10-x^{2}-y^{2}\) for \(-1 \leq x \leq 1,-3 \leq y \leq 3\) as a double integral. Evaluate the integral numerically using a CAS.
Evaluate \(\iint_{\mathcal{S}} x^{2} y d S\), where \(\mathcal{S}\) is the surface \(z=\sqrt{3} x+y^{2},-1 \leq x \leq 1,0 \leq y \leq 1\)
Calculate \(\iint_{\mathcal{S}}\left(x^{2}+y^{2}ight) e^{-z} d S\), where \(\mathcal{S}\) is the cylinder with equation \(x^{2}+y^{2}=9\) for \(0 \leq z \leq 10\).
Let SS be the upper hemisphere x2+y2+z2=1,z≥0x2+y2+z2=1,z≥0. For each of the functions (a)-(d), determine whether \(\iint_{\mathcal{S}} f d S\) is positive, zero, or negative (without evaluating
Let \(\mathcal{S}\) be a small patch of surface with a parametrization \(\Phi(u, v)\) for \(0 \leq u \leq 0.1,0 \leq v \leq 0.1\) such that the normal vector \(\mathbf{N}(u, v)\) for \((u, v)=(0,0)\)
The upper half of the sphere \(x^{2}+y^{2}+z^{2}=9\) has parametrization \(\Phi(r, \theta)=\left(r \cos \theta, r \sin \theta, \sqrt{9-r^{2}}ight)\) in cylindrical coordinates (Figure 3).(a)

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