a) A vector field F is called incompressible if div F = 0. Show that a vector field of the form F = <f(y,z),g(x,z),h(x,y)> is incompressible.
b) Suppose that S is a closed surface (a boundary of a solid in three dimensional space) and that F is an incompressible vector field. Show that the flux of F through S is 0.
c)Show that if f and g are defined on R3 and C is a closed curve
in R3 then 
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a) A vector field F is called incompressible if div F = 0. Show that a vector field of the form F = <f(y,z),g(x,z),h(x,y)> is incompressible. b) Suppose that S is a closed surface (a boundary o...
3. F is the vector field The surface S is the boundary of a solid E, where E is bounded by the sp...
3. F is the vector field The surface S is the boundary of a solid E, where E is bounded by the sphe:93 x2 + y2 + z2 = 4 and x2 + y2 + z2-) for z > 0, Do the following (a) State the defining equation for Gauss' Theorem. (10 points) (b) Show that div F(a+y). (10 points) (c) Use Gauss' Theorem to rewrite the following integral as product of one dimensional integrals. Do not evaluate. (10 points)...
Example A.3 Surface normal vector. Let S be a surface that is represented by f(x, y, z) -c, where...
Example A.3 Surface normal vector. Let S be a surface that is represented by f(x, y, z) -c, where f is defined and differentiable in a space. Then, let C be a curve on S through a point P-Go, yo,Zo) on S, where C is represented by rt)[x(t), y(t), z(t)] with r(to) -[xo. Vo, zol. Since C lies on S, r(t) must satisfy f(x, y. z)-c, or f(x(t), y(t), z(t))-c. Show that vf is orthogonal to any tangent vector r'(t)...
(1) Let F denote the inverse square vector field (axr, y, z) F= (Note that ||F...
(1) Let F denote the inverse square vector field (axr, y, z) F= (Note that ||F 1/r2.) The domain of F is R3\{(0, 0, 0)} where r = the chain rule (a) Verify that Hint: first show that then use (b) Show that div(F 0. (c) Suppose that S is a closed surface in R3 that does not enclose the origin. Show that the flux of F through S is zero. Hint: since the interior of S does not contain...
F·dS for the given vector field F and the oriented surface S. In other words, find the flux of F across S. For closed surfaces, use the positive (outward) Evaluate the surface integral orientation. F...
F·dS for the given vector field F and the oriented surface S. In other words, find the flux of F across S. For closed surfaces, use the positive (outward) Evaluate the surface integral orientation. F(x, y, z) -x2i +y^j+z2 k S is the boundary of the solid half-cylinder 0szs V 25 -y2, 0 sxs2 Need HelpRead It Watch Talk to a Tutor
F·dS for the given vector field F and the oriented surface S. In other words, find the flux...
Through what closed, oriented surface in R3 does the vector field F = < (4x +...
Through what closed, oriented surface in R3 does the vector
field F = < (4x + 2(x^3) z) , (−y(x^2 + z^2 ), −(3(x^2)(z^2) +
4(y^2)z) > have the greatest flux?
Through what closed, oriented surface in R3 does the vector field have the greatest flux?
(1) Let G(,y, z) = (x,y, z). Show that there exists no vector field A : R3 -> R3 such that curl(A) Hint: compute its...
(1) Let G(,y, z) = (x,y, z). Show that there exists no vector field A : R3 -> R3 such that curl(A) Hint: compute its divergence G. (2) Let H R3 -> R3 be given as H(x,y, z) = (1,2,3). Find a vector potential A : R3 -> R3 such that curl(A) smooth function = H. Show that if A is a vector potential for H, then so is A+ Vf, for any f : R5 -> R (3) Let...
Please answer without using previously posted answers. Thanks Let F(x, y) be a two-dimensional vector field. Spose further that there exists a scalar function, o, such that Then, F(x,y) is called a g...
Please answer without using previously posted answers.
Thanks
Let F(x, y) be a two-dimensional vector field. Spose further that there exists a scalar function, o, such that Then, F(x,y) is called a gradient field, and φ s called a potential function. Ideal Fluid Flow Let F represent the two-dimensional velocity field of an inviscid fluid that is incompressible, ie. . F-0 (or divergence-free). F can be represented by (1), where ф is called the velocity potential-show that o is harmonic....
Evaluate the surface integral F dS for the given vector field F and the oriented surface S. In other words, find the flux of F across S. For closed surfaces, use the positive (outward) orientation. F...
Evaluate the surface integral F dS for the given vector field F and the oriented surface S. In other words, find the flux of F across S. For closed surfaces, use the positive (outward) orientation. F(x, y, z) -xi yj+3 k S is the boundary of the region enclosed by the cylinder x2 + z2-1 and the planes y 0 and x y 2
Evaluate the surface integral F dS for the given vector field F and the oriented surface...
(1 point) Horizontal cross-sections of the vector fields F(x, y, z) and G(x, y, z) are...
(1 point) Horizontal cross-sections of the vector fields F(x, y, z) and G(x, y, z) are given in the figure. Each vector field has zero z-component (i.e., all of its vectors are horizontal) and is independent of 2 (i.e., is the same in every horizontal plane). You may assume that the graphs of these vector fields use the same scale. (a) Are div(F) and div(G) positive, negative, or zero at the origin? Be sure you can explain your answer. At...
(c) Let F be the vector field on R given by F(x, y, z) = (2x...
(c) Let F be the vector field on R given by F(x, y, z) = (2x +3y, z, 3y + z). (i) Calculate the divergence of F and the curl of F (ii) Let V be the region in IR enclosed by the plane I +2y +z S denote the closed surface that is the boundary of this region V. Sketch a picture of V and S. Then, using the Divergence Theorem, or otherwise, calculate 3 and the XY, YZ...
3. F is the vector field The surface S is the boundary of a solid E, where E is bounded by the sphe:93 x2 + y2 + z2 = 4 and x2 + y2 + z2-) for z > 0, Do the following (a) State the defining equation for Gauss' Theorem. (10 points) (b) Show that div F(a+y). (10 points) (c) Use Gauss' Theorem to rewrite the following integral as product of one dimensional integrals. Do not evaluate. (10 points)...
Example A.3 Surface normal vector. Let S be a surface that is represented by f(x, y, z) -c, where f is defined and differentiable in a space. Then, let C be a curve on S through a point P-Go, yo,Zo) on S, where C is represented by rt)[x(t), y(t), z(t)] with r(to) -[xo. Vo, zol. Since C lies on S, r(t) must satisfy f(x, y. z)-c, or f(x(t), y(t), z(t))-c. Show that vf is orthogonal to any tangent vector r'(t)...
(1) Let F denote the inverse square vector field (axr, y, z) F= (Note that ||F 1/r2.) The domain of F is R3\{(0, 0, 0)} where r = the chain rule (a) Verify that Hint: first show that then use (b) Show that div(F 0. (c) Suppose that S is a closed surface in R3 that does not enclose the origin. Show that the flux of F through S is zero. Hint: since the interior of S does not contain...
F·dS for the given vector field F and the oriented surface S. In other words, find the flux of F across S. For closed surfaces, use the positive (outward) Evaluate the surface integral orientation. F(x, y, z) -x2i +y^j+z2 k S is the boundary of the solid half-cylinder 0szs V 25 -y2, 0 sxs2 Need HelpRead It Watch Talk to a Tutor
F·dS for the given vector field F and the oriented surface S. In other words, find the flux...
Through what closed, oriented surface in R3 does the vector
field F = < (4x + 2(x^3) z) , (−y(x^2 + z^2 ), −(3(x^2)(z^2) +
4(y^2)z) > have the greatest flux?
Through what closed, oriented surface in R3 does the vector field have the greatest flux?
(1) Let G(,y, z) = (x,y, z). Show that there exists no vector field A : R3 -> R3 such that curl(A) Hint: compute its divergence G. (2) Let H R3 -> R3 be given as H(x,y, z) = (1,2,3). Find a vector potential A : R3 -> R3 such that curl(A) smooth function = H. Show that if A is a vector potential for H, then so is A+ Vf, for any f : R5 -> R (3) Let...
Please answer without using previously posted answers.
Thanks
Let F(x, y) be a two-dimensional vector field. Spose further that there exists a scalar function, o, such that Then, F(x,y) is called a gradient field, and φ s called a potential function. Ideal Fluid Flow Let F represent the two-dimensional velocity field of an inviscid fluid that is incompressible, ie. . F-0 (or divergence-free). F can be represented by (1), where ф is called the velocity potential-show that o is harmonic....
Evaluate the surface integral F dS for the given vector field F and the oriented surface S. In other words, find the flux of F across S. For closed surfaces, use the positive (outward) orientation. F(x, y, z) -xi yj+3 k S is the boundary of the region enclosed by the cylinder x2 + z2-1 and the planes y 0 and x y 2
Evaluate the surface integral F dS for the given vector field F and the oriented surface...
(1 point) Horizontal cross-sections of the vector fields F(x, y, z) and G(x, y, z) are given in the figure. Each vector field has zero z-component (i.e., all of its vectors are horizontal) and is independent of 2 (i.e., is the same in every horizontal plane). You may assume that the graphs of these vector fields use the same scale. (a) Are div(F) and div(G) positive, negative, or zero at the origin? Be sure you can explain your answer. At...
(c) Let F be the vector field on R given by F(x, y, z) = (2x +3y, z, 3y + z). (i) Calculate the divergence of F and the curl of F (ii) Let V be the region in IR enclosed by the plane I +2y +z S denote the closed surface that is the boundary of this region V. Sketch a picture of V and S. Then, using the Divergence Theorem, or otherwise, calculate 3 and the XY, YZ...
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