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mathematics
calculus 4th

Questions and Answers of Calculus 4th

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 y, x, e^{x z}ightangle, \quad x^{2}+y^{2}=9, x
Compute \(\iint_{\mathcal{S}} \mathbf{F} \cdot d \mathbf{S}\) for the given oriented surface or parametrized surface.\(\mathbf{F}(x, y, z)=\langle-y, z,-xangle, \quad \Phi(u, v)=(u+3 v, v-2 u, 2
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 0,0, x^{2}+y^{2}ightangle, \quad
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 0,0, x z e^{x y}ightangle, \quad z=x y, \quad 0
Compute \(\iint_{\mathcal{S}} \mathbf{F} \cdot d \mathbf{S}\) for the given oriented surface or parametrized surface.\(\mathbf{F}(x, y, z)=\langle 0,0, zangle, \quad 3 x^{2}+2 y^{2}+z^{2}=1, \quad z
Calculate the total charge on the cylinder\[x^{2}+y^{2}=R^{2}, \quad 0 \leq z \leq H\]if the charge density in cylindrical coordinates is \(\delta(\theta, z)=K z^{2} \cos ^{2} \theta\), where \(K\)
Find the flow rate of a fluid with velocity field \(\mathbf{v}=\langle 2 x, y, x yangle \mathrm{m} / \mathrm{s}\) across the part of the cylinder \(x^{2}+y^{2}=\) 9 where \(x \geq 0, y \geq 0\), and
With \(\mathbf{v}\) as in Exercise 59, calculate the flow rate across the part of the elliptic cylinder \(\frac{x^{2}}{4}+y^{2}=1\), where \(x \geq 0, y \geq 0\), and \(0 \leq z \leq 4\).Data From
Verify that \(\mathbf{F}=abla f\) and evaluate the line integral of \(\mathbf{F}\) over the given path.\(\mathbf{F}(x, y, z)=y e^{z} \mathbf{i}+x e^{z} \mathbf{j}+x y e^{z} \mathbf{k}, \quad f(x, y,
Let \(\mathcal{S}\) be the unit square in the \(x y\)-plane shown in Figure 14, oriented with the normal pointing in the positive \(z\)-direction. Estimate\[\iint_{\mathcal{S}} \mathbf{F} \cdot d
Compute the vector assigned to the point \(P=(-3,5)\) by the vector field:(a) \(\mathbf{F}(x, y)=\langle x y, y-xangle\)(b) \(\mathbf{F}(x, y)=\langle 4,8angle\)(c) \(\mathbf{F}(x, y)=\left\langle
Find a vector field \(\mathbf{F}\) in the plane such that \(\|\mathbf{F}(x, y)\|=1\) and \(\mathbf{F}(x, y)\) is orthogonal to \(\mathbf{G}(x, y)=\langle x, yangle\) for all \(x, y\).
Sketch the vector field.\(\mathbf{F}(x, y)=\langle y, 1angle\)
Sketch the vector field.\(\mathbf{F}(x, y)=\langle 4,1angle\)
Sketch the vector field.\(abla f\), where \(f(x, y)=x^{2}-y\)
Sketch the vector field.\(\mathbf{F}(x, y)=\left\langle\frac{4 y}{\sqrt{x^{2}+4 y^{2}}}, \frac{-x}{\sqrt{x^{2}+16 y^{2}}}ightangle\)Show that \(\mathbf{F}\) is a unit vector field tangent to the
Calculate \(\operatorname{div}(\mathbf{F})\) and \(\operatorname{curl}(\mathbf{F})\).\(\mathbf{F}=\left\langle x^{2}, y^{2}, z^{2}ightangle\)
Calculate \(\operatorname{div}(\mathbf{F})\) and \(\operatorname{curl}(\mathbf{F})\).\(\mathbf{F}=\langle y z, x z, x yangle\)
Calculate \(\operatorname{div}(\mathbf{F})\) and \(\operatorname{curl}(\mathbf{F})\).\(\mathbf{F}=\left\langle x^{3} y, x z^{2}, y^{2} zightangle\)
Calculate \(\operatorname{div}(\mathbf{F})\) and \(\operatorname{curl}(\mathbf{F})\).\(\mathbf{F}=\langle\sin x y, \cos y z, \sin x zangle\)
Calculate \(\operatorname{div}(\mathbf{F})\) and \(\operatorname{curl}(\mathbf{F})\).\(\mathbf{F}=y \mathbf{i}-z \mathbf{k}\)
\(\mathbf{F}(x, y)=\left\langle\frac{x+1}{(x+1)^{2}+y^{2}}, \frac{y}{(x+1)^{2}+y^{2}}ightangle ; \quad\) segment \(1 \leq y \leq 4\) along the \(y\)-axis, oriented upward
\(\mathbf{F}(x, y)=\left\langle e^{y}, 2 x-1ightangle\); parabola \(y=x^{2}\) for \(0 \leq x \leq 1\), oriented left to right
Let \(I=\int_{C} f(x, y, z) d s\). Assume that \(f(x, y, z) \geq m\) for some number \(m\) and all points \((x, y, z)\) on C. Which of the following conclusions is correct? Explain.(a) \(I \geq
Let \(\mathbf{F}(x, y)=\langle x, 0angle\). Prove that if \(C\) is any path from \((a, b)\) to \((c, d)\), then\[\int_{C} \mathbf{F} \cdot d \mathbf{r}=\frac{1}{2}\left(c^{2}-a^{2}ight)\]
Let \(\mathbf{F}(x, y)=\langle y, xangle\). Prove that if \(C\) is any path from \((a, b)\) to \((c, d)\), then\[\int_{C} \mathbf{F} \cdot d \mathbf{r}=c d-a b\]
We wish to define the average value Av(f)Av⁡(f) of a continuous function ff along a curve CC of length LL. Divide CC into NN consecutive arcs C1,…,CNC1,…,CN, each of length L/NL/N, and let PiPi
Use Eq. (10) to calculate the average value of \(f(x, y)=x-y\) along the segment from \(P=(2,1)\) to \(Q=(5,5)\). Av(5) = f(x. y. 2)ds L Je
Use Eq. (10) to calculate the average value of \(f(x, y)=x\) along the curve \(y=x^{2}\) for \(0 \leq x \leq 1\). solution The average value is Av(5)=f(x,y,z)ds L
The temperature (in degrees centigrade) at a point \(P\) on a circular wire of radius \(2 \mathrm{~cm}\) centered at the origin is equal to the square of the distance from \(P\) to \(P_{0}=(2,0)\).
The value of a scalar line integral does not depend on the choice of parametrization (because it is defined without reference to a parametrization). Prove this directly. That is, suppose that
The following statement is false. If \(\mathbf{F}\) is a gradient vector field, then the line integral of \(\mathbf{F}\) along every curve is zero. Which single word must be added to make it true?
Which of the following statements are true for all vector fields, and which are true only for conservative vector fields?(a) The line integral along a path from \(P\) to \(Q\) does not depend on
Let \(\mathbf{F}\) be a vector field on an open, connected domain \(\mathcal{D}\) with continuous second partial derivatives. Which of the following statements are always true, and which are true
Let \(C, D\), and \(\mathcal{E}\) be the oriented curves in Figure 16, and let \(\mathbf{F}=abla f\) be a gradient vector field such that \(\int_{C} \mathbf{F} \cdot d \mathbf{r}=4\). What are the
Let \(f(x, y, z)=x y \sin (y z)\) and \(\mathbf{F}=abla f\). Evaluate \(\int_{C} \mathbf{F} \cdot d \mathbf{r}\), where \(C\) is any path from \((0,0,0)\) to \((1,1, \pi)\).
Let \(\mathbf{F}(x, y, z)=\left\langle x^{-1} z, y^{-1} z, \ln (x y)ightangle\).(a) Verify that \(\mathbf{F}=abla f\), where \(f(x, y, z)=z \ln (x y)\).(b) Evaluate \(\int_{C} \mathbf{F} \cdot d
Verify that \(\mathbf{F}=abla f\) and evaluate the line integral of \(\mathbf{F}\) over the given path.\(\mathbf{F}(x, y)=\langle 3,6 yangle, \quad f(x, y)=3 x+3 y^{2} ; \quad
Verify that \(\mathbf{F}=abla f\) and evaluate the line integral of \(\mathbf{F}\) over the given path.\(\mathbf{F}(x, y)=\langle\cos y,-x \sin yangle, f(x, y)=x \cos y\); upper half of the unit
Verify that F=∇fF=∇f and evaluate the line integral of FF over the given path.\(\mathbf{F}(x, y, z)=\frac{z}{x} \mathbf{i}+\mathbf{j}+\ln x \mathbf{k}, \quad f(x, y, z)=y+z \ln x ;\)circle
Find a potential function for \(\mathbf{F}\) or determine that \(\mathbf{F}\) is not conservative.\(\mathbf{F}=\langle x, y, zangle\)
Find a potential function for \(\mathbf{F}\) or determine that \(\mathbf{F}\) is not conservative.\(\mathbf{F}=\langle y, x, zangle\)
Find a potential function for \(\mathbf{F}\) or determine that \(\mathbf{F}\) is not conservative.\(\mathbf{F}=\langle z, x, yangle\)
Find a potential function for \(\mathbf{F}\) or determine that \(\mathbf{F}\) is not conservative.\(\mathbf{F}=x \mathbf{j}+y \mathbf{k}\)
Find a potential function for F or determine that F is not conservative.\(\mathbf{F}=y^{2} \mathbf{i}+\left(2 x y+e^{z}ight) \mathbf{j}+y e^{z} \mathbf{k}\)
Find a potential function for F or determine that F is not conservative.\(\mathbf{F}=\left\langle y, x, z^{3}ightangle\)
Find a potential function for F or determine that F is not conservative.\(\mathbf{F}=\langle\cos (x z), \sin (y z), x y \sin zangle\)
Find a potential function for F or determine that F is not conservative.\(\mathbf{F}=\langle\cos z, 2 y,-x \sin zangle\)
Find a potential function for F or determine that F is not conservative.\(\mathbf{F}=\left\langle z \sec ^{2} x, z, y+\tan xightangle\)
Find a potential function for F or determine that F is not conservative.\(\mathbf{F}=\left\langle e^{x}(z+1),-\cos y, e^{x}ightangle\)
Find a potential function for F or determine that F is not conservative.\(\mathbf{F}=\left\langle 2 x y+5, x^{2}-4 z,-4 yightangle\)
Find a potential function for F or determine that F is not conservative.\(\mathbf{F}=\left\langle y z e^{x y}, x z e^{x y}-z, e^{x y}-yightangle\)
Evaluate\[\int_{C} 2 x y z d x+x^{2} z d y+x^{2} y d z\]over the path \(\mathbf{r}(t)=\left(t^{2}, \sin (\pi t / 4), e^{t^{2}-2 t}ight)\) for \(0 \leq t \leq 2\).
Evaluate\[\oint_{C} \sin x d x+z \cos y d y+\sin y d z\]where \(C\) is the ellipse \(4 x^{2}+9 y^{2}=36\), oriented clockwise.
Let \(\mathbf{F}=abla f\), and determine directly \(\int_{C} \mathbf{F} \cdot d \mathbf{r}\) for each of the two paths given, showing that they both give the same answer, which is
Let \(\mathbf{F}=abla f\), and determine directly \(\int_{C} \mathbf{F} \cdot d \mathbf{r}\) for each of the two paths given, showing that they both give the same answer, which is \(f(Q)-f(P)\).\(f=z
A vector field \(\mathbf{F}\) and contour lines of a potential function for \(\mathbf{F}\) are shown in Figure 17. Calculate the common value of \(\int_{C} \mathbf{F} \cdot d \mathbf{r}\) for the
Give a reason why the vector field \(\mathbf{F}\) in Figure 18 is not conservative. y 1 1 1 1 7 X
Calculate the work expended when a particle is moved from \(O\) to \(Q\) along segments \(\overline{O P}\) and \(\overline{P Q}\) in Figure 19 in the presence of the force field
Let \(\mathbf{F}(x, y)=\left\langle\frac{1}{x}, \frac{-1}{y}ightangle\). Calculate the work against \(F\) required to move an object from \((1,1)\) to \((3,4)\) along any path in the first quadrant.
Compute the work \(W\) against the earth's gravitational field required to move a satellite of mass \(m=1000\) \(\mathrm{kg}\) along any path from an orbit of altitude \(4000 \mathrm{~km}\) to an
An electric dipole with dipole moment \(p=4 \times 10^{-5} \mathrm{C}\)-m sets up an electric field (in newtons per coulomb)\[\mathbf{F}(x, y, z)=\frac{k p}{r^{5}}\left\langle 3 x z, 3 y z, 2
On the surface of the earth, the gravitational field (with \(z\) as vertical coordinate measured in meters) is \(\mathbf{F}=\langle 0,0,-gangle\).(a) Find a potential function for \(\mathbf{F}\).(b)
An electron at rest at \(P=(5,3,7)\) moves along a path ending at \(Q=(1,1,1)\) under the influence of the electric field (in newtons per coulomb)\[\mathbf{F}(x, y,
Let \(\mathbf{F}=\left\langle\frac{-y}{x^{2}+y^{2}}, \frac{x}{x^{2}+y^{2}}ightangle\) be the vortex field. Determine \(\int_{C} \mathbf{F} \cdot d \mathbf{r}\) for each of the paths in Figure 20. (A)
Show that \(g(x, y)=-\tan ^{-1} \frac{x}{y}\) is a potential function for the vortex field.
Determine whether or not the vector field \(\mathbf{F}(x, y)=\left\langle\frac{x^{2}}{x^{2}+y^{2}}, \frac{y^{2}}{x^{2}+y^{2}}ightangle\) has a potential function.
The vector field \(\mathbf{F}(x, y)=\left\langle\frac{x}{x^{2}+y^{2}}, \frac{y}{x^{2}+y^{2}}ightangle\) is defined on the domain \(\mathcal{D}=\{(x, y) eq(0,0)\}\).(a) Is \(\mathcal{D}\) simply
Suppose that \(\mathbf{F}\) is defined on \(\mathbf{R}^{3}\) and that \(\oint_{C} \mathbf{F} \cdot d \mathbf{r}=0\) for all closed paths \(C\) in \(\mathbf{R}^{3}\). Prove:(a) \(\mathbf{F}\) is path
What is the surface integral of the function \(f(x, y, z)=10\) over a surface of total area 5?
What interpretation can we give to the length \(\|\mathbf{N}\|\) of the normal vector for a parametrization \(\Phi(u, v)\) ?
A parametrization maps a rectangle of size \(0.01 \times 0.02\) in the \(u v\)-plane onto a small patch \(\mathcal{S}\) of a surface. Estimate \(\operatorname{area}(\mathcal{S})\) if \(\mathbf{T}_{u}
A small surface \(\mathcal{S}\) is divided into three small pieces, each of area 0.2 . Estimate \(\iint_{\mathcal{S}} f(x, y, z) d S\) if \(f(x, y, z)\) takes the values \(0.9,1\), and 1.1 at sample
A surface \(\mathcal{S}\) has a parametrization whose domain is the square \(0 \leq u, v \leq 2\) such that \(\|\mathbf{N}(u, v)\|=5\) for all \((u, v)\). What is area \((\mathcal{S})\) ?
What is the outward-pointing unit normal to the sphere of radius 3 centered at the origin at \(P=(2,2,1)\) ?
Match each parametrization with the corresponding surface in Figure 16.(a) \((u, \cos v, \sin v)\)(b) \((u, u+v, v)\)(c) \(\left(u, v^{3}, vight)\)(d) \((\cos u \sin v, 3 \cos u \sin v, \cos v)\)(e)
Show that \(\Phi(r, \theta)=\left(r \cos \theta, r \sin \theta, 1-r^{2}ight)\) parametrizes the paraboloid \(z=1-x^{2}-y^{2}\). Describe the grid curves of this parametrization.
Show that \(\Phi(u, v)=(2 u+1, u-v, 3 u+v)\) parametrizes the plane \(2 x-y-z=2\). Then(a) Calculate \(\mathbf{T}_{u}, \mathbf{T}_{v}\), and \(\mathbf{N}(u, v)\).(b) Find the area of
Let \(\mathcal{S}=\Phi(\mathcal{D})\), where \(\mathcal{D}=\left\{(u, v): u^{2}+v^{2} \leq 1, u \geq 0, v \geq 0ight\}\) and \(\Phi\) is as defined in Exercise 3.Data From Exercise 3Show that
Let \(\Phi(x, y)=(x, y, x y)\).(a) Calculate \(\mathbf{T}_{x}, \mathbf{T}_{y}\), and \(\mathbf{N}(x, y)\).(b) Let \(S\) be the part of the surface with parameter domain \(\mathcal{D}=\left\{(x, y):
A surface \(\mathcal{S}\) has a parametrization \(\Phi(u, v)\) whose domain \(\mathcal{D}\) is the square in Figure 17. Suppose that \(\Phi\) has the following normal
Calculate \(\mathbf{T}_{u}, \mathbf{T}_{v}\), and \(\mathbf{N}(u, v)\) for the parametrized surface at the given point. Then find the equation of the tangent plane to the surface at that
Calculate \(\mathbf{T}_{u}, \mathbf{T}_{v}\), and \(\mathbf{N}(u, v)\) for the parametrized surface at the given point. Then find the equation of the tangent plane to the surface at that
Calculate \(\mathbf{T}_{u}, \mathbf{T}_{v}\), and \(\mathbf{N}(u, v)\) for the parametrized surface at the given point. Then find the equation of the tangent plane to the surface at that
Calculate \(\mathbf{T}_{u}, \mathbf{T}_{v}\), and \(\mathbf{N}(u, v)\) for the parametrized surface at the given point. Then find the equation of the tangent plane to the surface at that
Use the normal vector computed in Exercise 8 to estimate the area of the small patch of the surface \(\Phi(u, v)=\left(u^{2}-v^{2}, u+v, u-vight)\) defined by\[2 \leq u \leq 2.1, \quad 3 \leq v \leq
Sketch the small patch of the sphere whose spherical coordinates satisfy\[\frac{\pi}{2}-0.15 \leq \theta \leq \frac{\pi}{2}+0.15, \quad \frac{\pi}{4}-0.1 \leq \phi \leq \frac{\pi}{4}+0.1\]Use the
Calculate \(\iint_{\mathcal{S}} f(x, y, z) d S\) for the given surface and function.\(\Phi(u, v)=(u \cos v, u \sin v, u), \quad 0 \leq u \leq 1, \quad 0 \leq v \leq 1 ; f(x, y,
Calculate \(\iint_{\mathcal{S}} f(x, y, z) d S\) for the given surface and function.\(\Phi(r, \theta)=(r \cos \theta, r \sin \theta, \theta), \quad 0 \leq r \leq 1, \quad 0 \leq \theta \leq 2 \pi ;
Calculate \(\iint_{\mathcal{S}} f(x, y, z) d S\) for the given surface and function.\(y=4-z^{2}, \quad 0 \leq x \leq 2,0 \leq z \leq 2 ; \quad f(x, y, z)=3 z\)
Calculate \(\iint_{\mathcal{S}} f(x, y, z) d S\) for the given surface and function.\(y=4-z^{2}, \quad 0 \leq x \leq z \leq 2 ; \quad f(x, y, z)=3\)
Calculate \(\iint_{\mathcal{S}} f(x, y, z) d S\) for the given surface and function.\(x^{2}+y^{2}+z^{2}=1, x, y, z \geq 0 ; \quad f(x, y, z)=x^{2}\)
Calculate \(\iint_{\mathcal{S}} f(x, y, z) d S\) for the given surface and function.\(z=4-x^{2}-y^{2}, \quad 0 \leq z \leq 3 ; \quad f(x, y, z)=x^{2} /(4-z)\)
Calculate \(\iint_{\mathcal{S}} f(x, y, z) d S\) for the given surface and function.\(x^{2}+y^{2}=4, \quad 0 \leq z \leq 4 ; \quad f(x, y, z)=e^{-z}\)
Calculate \(\iint_{\mathcal{S}} f(x, y, z) d S\) for the given surface and function.\(\Phi(u, v)=\left(u, v^{3}, u+vight), \quad 0 \leq u \leq 1,0 \leq v \leq 1\);\(f(x, y, z)=y\)
Calculate \(\iint_{\mathcal{S}} f(x, y, z) d S\) for the given surface and function.Part of the plane \(x+y+z=1\), where \(x, y, z \geq 0\);\(f(x, y, z)=z\)
Calculate \(\iint_{\mathcal{S}} f(x, y, z) d S\) for the given surface and function.Part of the plane \(x+y+z=0\) contained in the cylinder \(x^{2}+y^{2}=1 ; f(x, y, z)=z^{2}\)
Calculate \(\iint_{\mathcal{S}} f(x, y, z) d S\) for the given surface and function.\(x^{2}+y^{2}+z^{2}=4,1 \leq z \leq 2 ; \quad f(x, y, z)=z^{2}\left(x^{2}+y^{2}+z^{2}ight)^{-1}\)
Calculate \(\iint_{\mathcal{S}} f(x, y, z) d S\) for the given surface and function.\(x^{2}+y^{2}+z^{2}=4,0 \leq y \leq 1 ; \quad f(x, y, z)=y\)
Calculate \(\iint_{\mathcal{S}} f(x, y, z) d S\) for the given surface and function.Part of the surface \(z=x^{3}\), where \(0 \leq x \leq 1,0 \leq y \leq 1 ; \quad f(x, y, z)=z\)

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