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mathematics
calculus 4th
Calculus 4th Edition Jon Rogawski, Colin Adams, Robert Franzosa - Solutions
Use induction to prove that \(n !>2^{n}\) for \(n \geq 4\).Let \(\left\{F_{n}ight\}\) be the Fibonacci sequence, defined by the recursion formula\[F_{n}=F_{n-1}+F_{n-2}, \quad F_{1}=F_{2}=1\]
Use induction to prove the identity.\(F_{1}+F_{2}+\cdots+F_{n}=F_{n+2}-1\)
Use induction to prove the identity.\(F_{1}^{2}+F_{2}^{2}+\cdots+F_{n}^{2}=F_{n+1} F_{n}\)
Use induction to prove the identity.\(F_{n}=\frac{R_{+}^{n}-R_{-}^{n}}{\sqrt{5}}\), where \(R_{ \pm}=\frac{1 \pm \sqrt{5}}{2}\)
Use induction to prove the identity.\(F_{n+1} F_{n-1}=F_{n}^{2}+(-1)^{n}\). For the induction step, show that\[\begin{aligned}F_{n+2} F_{n} & =F_{n+1} F_{n}+F_{n}^{2} \\F_{n+1}^{2} & =F_{n+1} F_{n}+F_{n+1} F_{n-1}\end{aligned}\]
Use induction to prove that \(f(n)=8^{n}-1\) is divisible by 7 for all natural numbers \(n\). For the induction step, show that\[8^{k+1}-1=7 \cdot 8^{k}+\left(8^{k}-1ight)\]
Use induction to prove that \(n^{3}-n\) is divisible by 3 for all natural numbers \(n\).
Use induction to prove that \(5^{2 n}-4^{n}\) is divisible by 7 for all natural numbers \(n\).
Use Pascal's Triangle to write out the expansions of \((a+b)^{6}\) and \((a-b)^{4}\)
Expand the \(\left(x+x^{-1}ight)^{4}\).
What is the coefficient of \(x^{9}\) in \(\left(x^{3}+xight)^{5}\) ?
Let \(S(n)=\sum_{k=0}^{n}\left(\begin{array}{l}n \\ k\end{array}ight)\).(a) Use Pascal's Triangle to compute \(S(n)\) for \(n=1,2,3,4\).(b) Prove that \(S(n)=2^{n}\) for all \(n \geq 1\). Expand \((a+b)^{n}\) and evaluate at \(a=b=1\).
Let \(T(n)=\sum_{k=0}^{n}(-1)^{k}\left(\begin{array}{l}n \\ k\end{array}ight)\).(a) Use Pascal's Triangle to compute \(T(n)\) for \(n=1,2,3,4\).(b) Prove that \(T(n)=0\) for all \(n \geq 1\). Expand \((a+b)^{n}\) and evaluate at \(a=1, b=-1\).
Area of a Polygon Green's Theorem leads to a convenient formula for the area of a polygon.(a) Let \(C\) be the line segment joining \(\left(x_{1}, y_{1}ight)\) to \(\left(x_{2}, y_{2}ight)\). Show that\[\frac{1}{2} \int_{C}-y d x+x d y=\frac{1}{2}\left(x_{1} y_{2}-x_{2} y_{1}ight)\](b) Prove that
Use the result of Exercise 36 to compute the areas of the polygons in Figure 28. Check your result for the area of the triangle in \((\mathrm{A})\) using geometry.Data From Exercise 36Area of a Polygon Green's Theorem leads to a convenient formula for the area of a polygon.(a) Let \(C\) be the line
Compute the flux \(\oint \mathbf{F} \cdot \mathbf{n} d\) s of \(\mathbf{F}\) across the curve \(C\) for the given vector field and curve using the vector form of Green's Theorem.\(\mathbf{F}(x, y)=\langle 3 x, 2 yangle\) across the circle given by \(x^{2}+y^{2}=9\) THEOREM 1 Green's Theorem Let D
Compute the flux \(\oint \mathbf{F} \cdot \mathbf{n} d\) s of \(\mathbf{F}\) across the curve \(C\) for the given vector field and curve using the vector form of Green's Theorem.\(\mathbf{F}(x, y)=\langle x y, x-yangle\) across the boundary of the square \(-1 \leq x \leq 1,-1 \leq y \leq 1\)
Compute the flux \(\oint \mathbf{F} \cdot \mathbf{n} d\) s of \(\mathbf{F}\) across the curve \(C\) for the given vector field and curve using the vector form of Green's Theorem.\(\mathbf{F}(x, y)=\left\langle x^{2}, y^{2}ightangle\) across the boundary of the triangle with vertices
Compute the flux \(\oint \mathbf{F} \cdot \mathbf{n} d\) s of \(\mathbf{F}\) across the curve \(C\) for the given vector field and curve using the vector form of Green's Theorem.\(\mathbf{F}(x, y)=\left\langle x^{2}, y^{2}ightangle\) across the boundary of the triangle with vertices
Compute the flux \(\oint \mathbf{F} \cdot \mathbf{n} d\) s of \(\mathbf{F}\) across the curve \(C\) for the given vector field and curve using the vector form of Green's Theorem.\(\mathbf{F}(x, y)=\langle\cos y, \sin yangle\) across the boundary of the square \(0 \leq x \leq 2,0 \leq y \leq
Compute the flux \(\oint \mathbf{F} \cdot \mathbf{n} d\) s of \(\mathbf{F}\) across the curve \(C\) for the given vector field and curve using the vector form of Green's Theorem.\(\mathbf{F}(x, y)=\left\langle x y^{2}+2 x, x^{2} y-2 yightangle\) across the simple closed curve that is the boundary
If \(\mathbf{v}\) is the velocity field of a fluid, the flux of \(\mathbf{v}\) across \(C\) is equal to the flow rate (amount of fluid flowing across \(C\) in square meters per second). Find the flow rate across the circle of radius 2 centered at the origin if
A buffalo stampede (Figure 29) is described by a velocity vector field F = (xy — y3, x2 + y) kilometers per hour in the region 7, defined by 2 THEOREM 1 Green's Theorem Let D be a domain whose boundary 3D is a simple closed curve, oriented counterclockwise. If F and F2 have continuous partial
The Laplace operator \(\Delta\) is defined by\[\Delta \varphi=\frac{\partial^{2} \varphi}{\partial x^{2}}+\frac{\partial^{2} \varphi}{\partial y^{2}}\]For any vector field \(\mathbf{F}=\left\langle F_{1}, F_{2}ightangle\), define the conjugate vector field \(\mathbf{F}^{*}=\left\langle-F_{2},
The Laplace operator \(\Delta\) is defined by\[\Delta \varphi=\frac{\partial^{2} \varphi}{\partial x^{2}}+\frac{\partial^{2} \varphi}{\partial y^{2}}\]For any vector field \(\mathbf{F}=\left\langle F_{1}, F_{2}ightangle\), define the conjugate vector field \(\mathbf{F}^{*}=\left\langle-F_{2},
The Laplace operator \(\Delta\) is defined by\[\Delta \varphi=\frac{\partial^{2} \varphi}{\partial x^{2}}+\frac{\partial^{2} \varphi}{\partial y^{2}}\]For any vector field \(\mathbf{F}=\left\langle F_{1}, F_{2}ightangle\), define the conjugate vector field \(\mathbf{F}^{*}=\left\langle-F_{2},
The Laplace operator \(\Delta\) is defined by\[\Delta \varphi=\frac{\partial^{2} \varphi}{\partial x^{2}}+\frac{\partial^{2} \varphi}{\partial y^{2}}\]For any vector field \(\mathbf{F}=\left\langle F_{1}, F_{2}ightangle\), define the conjugate vector field \(\mathbf{F}^{*}=\left\langle-F_{2},
Let \(\mathcal{D}\) be the region bounded by a simple closed curve \(C\). A function \(\varphi(x, y)\) on \(\mathcal{D}\) (whose second-order partial derivatives exist and are continuous) is called harmonic if \(\Delta \varphi=0\), where \(\Delta \varphi\) is the Laplace operator defined in
Let \(\mathcal{D}\) be the region bounded by a simple closed curve \(C\). A function \(\varphi(x, y)\) on \(\mathcal{D}\) (whose second-order partial derivatives exist and are continuous) is called harmonic if \(\Delta \varphi=0\), where \(\Delta \varphi\) is the Laplace operator defined in
What is the definition of a vector potential?
Which of the following statements is correct?(a) The flux of \(\operatorname{curl}(\mathbf{A})\) through every oriented surface is zero.(b) The flux of \(\operatorname{curl}(\mathbf{A})\) through every closed, oriented surface is zero.
Which condition on \(\mathbf{F}\) guarantees that the flux through \(\mathcal{S}_{1}\) is equal to the flux through \(\mathcal{S}_{2}\) for any two oriented surfaces \(\mathcal{S}_{1}\) and \(\mathcal{S}_{2}\) with the same oriented boundary?
Verify Stokes' Theorem for the given vector field and surface, oriented with an upwardpointing normal.\(\mathbf{F}=\langle 2 x y, x, y+zangle\), the surface \(z=1-x^{2}-y^{2}\) for \(x^{2}+y^{2} \leq 1\) THEOREM 1 Stokes' Theorem Let S be a surface as described earlier, and let F be a vector field
Verify Stokes' Theorem for the given vector field and surface, oriented with an upwardpointing normal.\(\mathbf{F}=\langle y z, 0, xangle, \quad\) the portion of the plane \(\frac{x}{2}+\frac{y}{3}+z=1\), where \(x, y, z \geq 0\) THEOREM 1 Stokes' Theorem Let S be a surface as described earlier,
Verify Stokes' Theorem for the given vector field and surface, oriented with an upwardpointing normal.\(\mathbf{F}=\left\langle e^{y-z}, 0,0ightangle\), the square with vertices \((1,0,1),(1,1,1),(0,1,1)\), and \((0,0,1)\) THEOREM 1 Stokes' Theorem Let S be a surface as described earlier, and let F
Verify Stokes' Theorem for the given vector field and surface, oriented with an upwardpointing normal.\(\mathbf{F}=\left\langle y, x, x^{2}+y^{2}ightangle\), the upper hemisphere \(x^{2}+y^{2}+z^{2}=1, z \geq 0\) THEOREM 1 Stokes' Theorem Let S be a surface as described earlier, and let F be a
Calculate curl(F) and then apply Stokes' Theorem to compute the flux of \(\operatorname{curl}(\mathbf{F})\) through the given surface using a line integral.\(\mathbf{F}=\left\langle e^{z^{2}}-y, e^{z^{3}}+x, \cos (x z)ightangle\), the upper half of the unit sphere \(x^{2}+y^{2}+z^{2}=1, z \geq 0\)
Calculate curl(F) and then apply Stokes' Theorem to compute the flux of curl(F)curl(F) through the given surface using a line integral.\(\left.\mathbf{F}=\left\langle x+y, z^{2}-4, x \sqrt{y^{2}+1}ight)ightangle\), surface of the wedge-shaped box in Figure 15 (bottom included, top excluded) with
Calculate curl(F) and then apply Stokes' Theorem to compute the flux of \(\operatorname{curl}(\mathbf{F})\) through the given surface using a line integral.\(\mathbf{F}=\langle 3 z, 5 x,-2 yangle\), that part of the paraboloid \(z=x^{2}+y^{2}\) that lies below the plane \(z=4\) with upwardpointing
Calculate curl(F) and then apply Stokes' Theorem to compute the flux of \(\operatorname{curl}(\mathbf{F})\) through the given surface using a line integral.\(\mathbf{F}=\left\langle y z,-x z, z^{3}ightangle\), that part of the cone \(z=\sqrt{x^{2}+y^{2}}\) that lies between the two planes \(z=1\)
Calculate curl(F) and then apply Stokes' Theorem to compute the flux of \(\operatorname{curl}(\mathbf{F})\) through the given surface using a line integral.\(\mathbf{F}=\langle y z, x z, x yangle\), that part of the cylinder \(x^{2}+y^{2}=1\) that lies between the two planes \(z=1\) and \(z=4\)
Calculate curl(F) and then apply Stokes' Theorem to compute the flux of \(\operatorname{curl}(\mathbf{F})\) through the given surface using a line integral.\(\mathbf{F}=\left\langle 2 y, e^{z},-\arctan xightangle\), that part of the paraboloid \(z=4-x^{2}-y^{2}\) cut off by the \(x y\)-plane with
Apply Stokes' Theorem to evaluate \(\oint_{C} \mathbf{F} \cdot d \mathbf{r}\) by finding the flux of \(\operatorname{curl}(\mathbf{F})\) across an appropriate surface.\(\mathbf{F}=\langle 3 y,-2 x, 3 yangle, \quad C\) is the circle \(x^{2}+y^{2}=9, z=2\), oriented counterclockwise as viewed from
Apply Stokes' Theorem to evaluate \(\oint_{C} \mathbf{F} \cdot d \mathbf{r}\) by finding the flux of \(\operatorname{curl}(\mathbf{F})\) across an appropriate surface.\(\mathbf{F}=\langle y z, x y, x zangle, \quad C\) is the square with vertices \((0,0,2),(1,0,2),(1,1,2)\), and \((0,1,2)\),
Apply Stokes' Theorem to evaluate \(\oint_{C} \mathbf{F} \cdot d \mathbf{r}\) by finding the flux of \(\operatorname{curl}(\mathbf{F})\) across an appropriate surface.\(\mathbf{F}=\langle x z, x y, y zangle, \quad C\) is the rectangle with vertices \((0,0,0),(0,0,2),(3,0,2)\), and \((3,0,0)\),
Apply Stokes' Theorem to evaluate \(\oint_{C} \mathbf{F} \cdot d \mathbf{r}\) by finding the flux of \(\operatorname{curl}(\mathbf{F})\) across an appropriate surface.\(\mathbf{F}=\langle y+2 x, 2 x+5 z, 7 y+8 xangle, \quad C\) is the circle with radius 5 , center at \((2,0,0)\), in the plane
Apply Stokes' Theorem to evaluate \(\oint_{C} \mathbf{F} \cdot d \mathbf{r}\) by finding the flux of \(\operatorname{curl}(\mathbf{F})\) across an appropriate surface.\(\mathbf{F}=\langle y, z, xangle, \quad C\) is the triangle with vertices \((0,0,0),(3,0,0)\), and \((0,3,3)\), oriented
Apply Stokes' Theorem to evaluate \(\oint_{C} \mathbf{F} \cdot d \mathbf{r}\) by finding the flux of \(\operatorname{curl}(\mathbf{F})\) across an appropriate surface.\(\mathbf{F}=\langle y,-2 z, 4 xangle, \quad C\) is the boundary of that portion of the plane \(x+2 y+3 z=1\) that is in the first
Let \(\mathcal{S}\) be the surface of the cylinder (not including the top and bottom) of radius 2 for \(1 \leq z \leq 6\), oriented with outward-pointing normal (Figure 16).(a) Indicate with an arrow the orientation of \(\partial \mathcal{S}\) (the top and bottom circles).(b) Verify Stokes' Theorem
Let \(\mathcal{S}\) be the portion of the plane \(z=x\) contained in the half-cylinder of radius \(R\) depicted in Figure 17. Use Stokes' Theorem to calculate the circulation of \(\mathbf{F}=\langle z, x, y+2 zangle\) around the boundary of \(\mathcal{S}\) (a halfellipse) in the counterclockwise
Let \(I\) be the flux of \(\mathbf{F}=\left\langle e^{y}, 2 x e^{x^{2}}, z^{2}ightangle\) through the upper hemisphere \(\mathcal{S}\) of the unit sphere.(a) Let \(\mathbf{G}=\left\langle e^{y}, 2 x e^{x^{2}}, 0ightangle\). Find a vector field \(\mathbf{A}\) such that
Let \(\mathbf{F}=\langle 0,-z, 1angle\). Let \(\mathcal{S}\) be the spherical cap \(x^{2}+y^{2}+z^{2} \leq 1\), where \(z \geq \frac{1}{2}\). Evaluate \(\iint_{\mathcal{S}} \mathbf{F} \cdot d \mathbf{S}\) directly as a surface integral. Then verify that
Let \(\mathbf{A}\) be the vector potential and \(\mathbf{B}\) the magnetic field of the infinite solenoid of radius \(R\) in Example 4 . Use Stokes' Theorem to compute:(a) The flux of \(\mathbf{B}\) through a circle in the \(x y\)-plane of radius \(r(b) The circulation of \(\mathbf{A}\) around the
The magnetic field \(\mathbf{B}\) due to a small current loop (which we place at the origin) is called a magnetic dipole (Figure 18). For \(ho\) large, \(\mathbf{B}=\operatorname{curl}(\mathbf{A})\), where\[\mathbf{A}=\left\langle-\frac{y}{ho^{3}}, \frac{x}{ho^{3}}, 0ightangle \text { and }
A uniform magnetic field \(\mathbf{B}\) has constant strength \(b\) in the \(z\)-direction [i.e., \(\mathbf{B}=\langle 0,0, bangle\) ].(a) Verify that \(\mathbf{A}=\frac{1}{2} \mathbf{B} \times \mathbf{r}\) is a vector potential for \(\mathbf{B}\), where \(\mathbf{r}=\langle x, y, 0angle\).(b)
Let \(\mathbf{F}=\left\langle-x^{2} y, x, 0ightangle\). Referring to Figure 19 , let \(C\) be the closed path \(A B C D\). Use Stokes' Theorem to evaluate \(\int_{C} \mathbf{F} \cdot d \mathbf{r}\) in two ways. First, regard \(C\) as the boundary of the rectangle with vertices \(A, B, C\), and
Let \(\mathbf{F}=\left\langle y^{2}, 2 z+x, 2 y^{2}ightangle\). Use Stokes' Theorem to find a plane with equation \(a x+b y+c z=0\) (where \(a, b, c\) are not all zero) such that \(\oint_{C} \mathbf{F} \cdot d \mathbf{r}=0\) for every closed \(C\) lying in the plane. Hint: Choose \(a, b, c\) so
Let \(\mathbf{F}=\left\langle-z^{2}, 2 z x, 4 y-x^{2}ightangle\), and let \(C\) be a simple closed curve in the plane \(x+y+z=4\) that encloses a region of area 16 (Figure 20). Calculate \(\oint_{C} \mathbf{F} \cdot d \mathbf{r}\), where \(C\) is oriented in the counterclockwise direction (when
Let \(\mathbf{F}=\left\langle y^{2}, x^{2}, z^{2}ightangle\). Show that\[\int_{C_{1}} \mathbf{F} \cdot d \mathbf{r}=\int_{C_{2}} \mathbf{F} \cdot d \mathbf{r}\]for any two closed curves going around a cylinder whose central axis is the z-axis as shown in Figure 21. N C C
The curl of a vector field \(\mathbf{F}\) at the origin is \(\mathbf{v}_{0}=\langle 3,1,4angle\). Estimate the circulation around the small parallelogram spanned by the vectors \(\mathbf{A}=\left\langle 0, \frac{1}{2}, \frac{1}{2}ightangle\) and \(\mathbf{B}=\left\langle 0,0, \frac{1}{3}ightangle\).
You know two things about a vector field \(\mathbf{F}\) :(i) \(\mathbf{F}\) has a vector potential \(\mathbf{A}\) (but \(\mathbf{A}\) is unknown).(ii) The circulation of \(\mathbf{A}\) around the unit circle (oriented counterclockwise) is 25 .Determine the flux of \(\mathbf{F}\) through the surface
Suppose that \(\mathbf{F}\) has a vector potential and that \(\mathbf{F}(x, y, 0)=\mathbf{k}\). Find the flux of \(\mathbf{F}\) through the surface \(\mathcal{S}\) in Figure 22, oriented with an upward-pointing normal. S N Unit circle
Prove that \(\operatorname{curl}(f \mathbf{a})=abla f \times \mathbf{a}\), where \(f\) is a differentiable function and \(\mathbf{a}\) is a constant vector.
Show that \(\operatorname{curl}(\mathbf{F})=\mathbf{0}\) if \(\mathbf{F}\) is radial, meaning that \(\mathbf{F}=f(ho)\langle x, y, zangle\) for some function \(f(ho)\), where \(ho=\sqrt{x^{2}+y^{2}+z^{2}}\). Hint: It is enough to show that one component of \(\operatorname{curl}(\mathbf{F})\) is
Prove the following Product Rule:\[\operatorname{curl}(f \mathbf{F})=f \operatorname{curl}(\mathbf{F})+abla f \times \mathbf{F}\]
Assume that \(f\) and \(g\) have continuous partial derivatives of order 2. Prove that\[\oint_{\partial \mathcal{S}} f abla g \cdot d \mathbf{r}=\iint_{\mathcal{S}} abla f \times abla g \cdot d \mathbf{S}\]
Verify that \(\mathbf{B}=\operatorname{curl}(\mathbf{A})\) for \(r>R\) in the setting of Example 4. EXAMPLE 4 Vector Potential for a Solenoid An electric current I flowing through a solenoid (a tightly wound spiral of wire; see Figure 11) creates a magnetic field B. If we assume that the
Explain carefully why Green's Theorem is a special case of Stokes' Theorem. THEOREM 1 Stokes' Theorem Let S be a surface as described earlier, and let F be a vector field whose components have continuous partial derivatives on an open region containing S. = [[ k F.dr= curl(F). dS The integral on
In this exercise, we use the notation of the proof of Theorem 1 and prove\[\oint_{C} F_{3}(x, y, z) \mathbf{k} \cdot d \mathbf{r}=\iint_{\mathcal{S}} \operatorname{curl}\left(F_{3}(x, y, z) \mathbf{k}ight) \cdot d \mathbf{S}\]In particular, \(\mathcal{S}\) is the graph of \(z=f(x, y)\) over a
Let \(\mathbf{F}\) be a continuously differentiable vector field in \(\mathbf{R}^{3}, Q\) a point, and \(\mathcal{S}\) a plane containing \(Q\) with unit normal vector e. Let \(C_{r}\) be a circle of radius \(r\) centered at \(Q\) in \(\mathcal{S}\), and let \(\mathcal{S}_{r}\) be the disk enclosed
What is the flux of \(\mathbf{F}=\langle 1,0,0angle\) through a closed surface?
Justify the following statement: The flux of \(\mathbf{F}=\left\langle x^{3}, y^{3}, z^{3}ightangle\) through every closed surface is positive.
Which of the following expressions are meaningful (where \(\mathbf{F}\) is a vector field and \(f\) is a function)? Of those that are meaningful, which are automatically zero?(a) \(\operatorname{div}(abla f)\)(b) \(\operatorname{curl}(abla f)\)(c) \(abla \operatorname{curl}(f)\)(d)
Which of the following statements is correct (where \(\mathbf{F}\) is a continuously differentiable vector field defined everywhere)?(a) The flux of \(\operatorname{curl}(\mathbf{F})\) through all surfaces is zero.(b) If \(\mathbf{F}=abla \varphi\), then the flux of \(\mathbf{F}\) through all
How does the Divergence Theorem imply that the flux of the vector field \(\mathbf{F}=\left\langle x^{2}, y-e^{z}, y-2 z xightangle\) through a closed surface is equal to the enclosed volume?
Verify the Divergence Theorem for the vector field and region.\(\mathbf{F}(x, y, z)=\langle z, x, yangle\), the box \([0,4] \times[0,2] \times[0,3]\) THEOREM 1 Divergence Theorem Let S be a closed surface that encloses a region W in R. Assume that S is piecewise smooth and is oriented by normal
Verify the Divergence Theorem for the vector field and region.\(\mathbf{F}(x, y, z)=\langle y, x, zangle\), the region \(x^{2}+y^{2}+z^{2} \leq 4\) THEOREM 1 Divergence Theorem Let S be a closed surface that encloses a region W in R. Assume that S is piecewise smooth and is oriented by normal
Verify the Divergence Theorem for the vector field and region.\(\mathbf{F}(x, y, z)=\langle 2 x, 3 z, 3 yangle\), the region \(x^{2}+y^{2} \leq 1,0 \leq z \leq 2\) THEOREM 1 Divergence Theorem Let S be a closed surface that encloses a region W in R. Assume that S is piecewise smooth and is oriented
Verify the Divergence Theorem for the vector field and region.\(\mathbf{F}(x, y, z)=\langle x, 0,0angle\), the region \(x^{2}+y^{2} \leq z \leq 4\) THEOREM 1 Divergence Theorem Let S be a closed surface that encloses a region W in R. Assume that S is piecewise smooth and is oriented by normal
Use the Divergence Theorem to evaluate the flux \(\iint_{\mathcal{S}} \mathbf{F} \cdot d \mathbf{S}\).\(\mathbf{F}(x, y, z)=\left\langle 0,0, z^{3} / 3ightangle, \mathcal{S}\) is the sphere \(x^{2}+y^{2}+z^{2}=1\). THEOREM 1 Divergence Theorem Let S be a closed surface that encloses a region W in
Use the Divergence Theorem to evaluate the flux \(\iint_{\mathcal{S}} \mathbf{F} \cdot d \mathbf{S}\).\(\mathbf{F}(x, y, z)=\langle y, z, xangle, \mathcal{S}\) is the sphere \(x^{2}+y^{2}+z^{2}=1\). THEOREM 1 Divergence Theorem Let S be a closed surface that encloses a region W in R. Assume that S
Use the Divergence Theorem to evaluate the flux \(\iint_{\mathcal{S}} \mathbf{F} \cdot d \mathbf{S}\).\(\mathbf{F}(x, y, z)=\left\langle x y^{2}, y z^{2}, z x^{2}ightangle, \mathcal{S}\) is the boundary of the cylinder given by \(x^{2}+y^{2} \leq 4,0 \leq z \leq 3\). THEOREM 1 Divergence Theorem
Use the Divergence Theorem to evaluate the flux \(\iint_{\mathcal{S}} \mathbf{F} \cdot d \mathbf{S}\).\(\mathbf{F}(x, y, z)=\left\langle x^{2} z, y x, x y zightangle, \mathcal{S}\) is the boundary of the tetrahedron given by \(x+y+z \leq 1,0 \leq x, 0 \leq y, 0 \leq z\). THEOREM 1 Divergence
Use the Divergence Theorem to evaluate the flux \(\iint_{\mathcal{S}} \mathbf{F} \cdot d \mathbf{S}\).\(\mathbf{F}(x, y, z)=\left\langle x+z^{2}, x z+y^{2}, z x-yightangle, \mathcal{S}\) is the surface that bounds the solid region with boundary given by the parabolic cylinder \(z=1-x^{2}\), and the
Use the Divergence Theorem to evaluate the flux \(\iint_{\mathcal{S}} \mathbf{F} \cdot d \mathbf{S}\).\(\mathbf{F}(x, y, z)=\left\langle z x, y x^{3}, x^{2} zightangle, \mathcal{S}\) is the surface that bounds the solid region with boundary given by \(y=\) \(4-x^{2}-z^{2}, y=0\). THEOREM 1
Use the Divergence Theorem to evaluate the flux \(\iint_{\mathcal{S}} \mathbf{F} \cdot d \mathbf{S}\).\(\mathbf{F}(x, y, z)=\left\langle x^{3}, 0, z^{3}ightangle, \mathcal{S}\) is the boundary of the region in the first octant of space given by \(x^{2}+y^{2}+z^{2} \leq 4\), \(x \geq 0, y \geq 0, z
Use the Divergence Theorem to evaluate the flux \(\iint_{\mathcal{S}} \mathbf{F} \cdot d \mathbf{S}\).\(\mathbf{F}(x, y, z)=\left\langle e^{x+y}, e^{x+z}, e^{x+y}ightangle, \mathcal{S}\) is the boundary of the unit cube \(0 \leq x \leq 1,0 \leq y \leq 1,0 \leq z \leq 1\). THEOREM 1 Divergence
Use the Divergence Theorem to evaluate the flux \(\iint_{\mathcal{S}} \mathbf{F} \cdot d \mathbf{S}\).\(\mathbf{F}(x, y, z)=\left\langle x, y^{2}, z+yightangle, \mathcal{S}\) is the boundary of the region contained in the cylinder \(x^{2}+y^{2}=4\) between the planes \(z=x\) and \(z=8\). THEOREM 1
Use the Divergence Theorem to evaluate the flux \(\iint_{\mathcal{S}} \mathbf{F} \cdot d \mathbf{S}\).\(\mathbf{F}(x, y, z)=\left\langle x^{2}-z^{2}, e^{z^{2}}-\cos x, y^{3}ightangle, \mathcal{S}\) is the boundary of the region bounded by \(x+2 y+4 z=12\) and the coordinate planes in the first
Use the Divergence Theorem to evaluate the flux \(\iint_{\mathcal{S}} \mathbf{F} \cdot d \mathbf{S}\).\(\mathbf{F}(x, y, z)=\langle x+y, z, z-xangle, \mathcal{S}\) is the boundary of the region between the paraboloid \(z=9-x^{2}-y^{2}\) and the \(x y\)-plane. THEOREM 1 Divergence Theorem Let S be a
Use the Divergence Theorem to evaluate the flux \(\iint_{\mathcal{S}} \mathbf{F} \cdot d \mathbf{S}\).\(\mathbf{F}(x, y, z)=\left\langle e^{z^{2}}, 2 y+\sin \left(x^{2} zight), 4 z+\sqrt{x^{2}+9 y^{2}}ightangle, \mathcal{S}\) is the region \(x^{2}+y^{2} \leq z \leq 8-x^{2}-y^{2}\). THEOREM 1
For the arch and loading in Figure P6.18, compute the reactions and determine the height of each point. The maximum height permitted at any point along the arch, \(h_{z \max }\), is \(20 \mathrm{~m}\).
Let \(\mathcal{S}_{1}\) be the closed surface consisting of \(\mathcal{S}\) in Figure 18 together with the unit disk. Find the volume enclosed by \(\mathcal{S}_{1}\), assuming that\[\iint_{\mathcal{S}_{1}}\langle x, 2 y, 3 zangle \cdot d \mathbf{S}=72\] S X 1 Unit circle
Let \(\mathcal{S}\) be the half-cylinder \(x^{2}+y^{2}=1, x \geq 0,0 \leq z \leq 1\). Assume that \(\mathbf{F}\) is a horizontal vector field (the \(z\)-component is zero) such that \(\mathbf{F}(0, y, z)=z y^{2} \mathbf{i}\). Let \(\mathcal{W}\) be the solid region enclosed by \(\mathcal{S}\), and
Volume as a Surface Integral Let \(\mathbf{F}(x, y, z)=\langle x, y, zangle\). Prove that if \(\mathcal{W}\) is a region in \(\mathbf{R}^{3}\) with a smooth boundary \(\mathcal{S}\), then\[\operatorname{volume}(\mathcal{W})=\frac{1}{3} \iint_{\mathcal{S}} \mathbf{F} \cdot d \mathbf{S}\]
Use Eq. (9) to calculate the volume of the unit ball as a surface integral over the unit sphere.Eq. (9) volume(W) 3. JfF.ds dS S
Verify that Eq. (9) applied to the box \([0, a] \times[0, b] \times[0, c]\) yields the volume \(V=a b c\).
Let WW be the region in Figure 19 bounded by the cylinder x2+y2=4x2+y2=4, the plane z=x+1z=x+1, and the xyxy-plane. Use the Divergence Theorem to compute the flux of F(x,y,z)=⟨z,x,y+z2⟩F(x,y,z)=⟨z,x,y+z2⟩ through the boundary of WW.
Let \(I=\iint_{\mathcal{S}} \mathbf{F} \cdot d \mathbf{S}\), where\[\mathbf{F}(x, y, z)=\left\langle\frac{2 y z}{r^{2}},-\frac{x z}{r^{2}},-\frac{x y}{r^{2}}ightangle\]\(\left(r=\sqrt{x^{2}+y^{2}+z^{2}}ight)\) and \(\mathcal{S}\) is the boundary of a region \(\mathcal{W}\).(a) Check that
The velocity field of a fluid \(\mathbf{v}\) (in meters per second) has divergence \(\operatorname{div}(\mathbf{v})(P)=3\) at the point \(P=\) \((2,2,2)\). Estimate the flow rate out of the sphere of radius 0.5 centered at \(P\).
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