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6. (20pt) Verify Stokes's theorem in the following example by doing the line integral in (a) and the flux integral in (b). (a) Evaluate the line integral of the vector field F [y+2,2+ x,x+y] along the boundary curve of the surface: 2-1-x2-y, z2-1. (b) Evaluate the flux integral of Vx F through the surface mentioned in (a) and compare the result with the result obtained in (a).
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Notation and convention: r x +y The distance from the origin to the point r [x,y,z]...
first picture is the notation on the test paper Second picture is the question i would...
first picture is the notation on the test paper
Second picture is the question i would like to solve
thanks~
Notation and convention: r x +y The distance from the origin to the point r [x,y,z] + ê: The unit vector along the direction of r-[x, y,z] (a.e,6)-i.j.):m :The orthonormal bases of a Cartesian coordinate system. for dummy indices Einstein convention: Omitting the summation notation (repeated indices). Examples:ab,-a b, ab a b Notice: No dummy index is allowed to be...
5. A surface is parameterized by u and v as xacosucosv y bsinucosv z csinv Here...
5. A surface is parameterized by u and v as xacosucosv y bsinucosv z csinv Here 0su<2r, -/2<v< /2, a, b and c are nonzero constants. (a) Express the surface equation in the form f(x, y,z) 1. Find f(x,y,2) (b) Calculate the gradient of the f mentioned above and find the unit normal vector n ar ar (c) Calculate the (un-normalized) normal vector N=- X Cu xr, and normalize it to get the unit normal vector n again. Notation and...
(1) Let P denote the solid bounded by the surface of the hemisphere z -Vl-r-y? and the cone2y2 and let n denote an outw...
(1) Let P denote the solid bounded by the surface of the hemisphere z -Vl-r-y? and the cone2y2 and let n denote an outwardly directed unit normal vector. Define the vector field F(x, y, z) = yi + zVJ + 21k. (a) Evaluate the surface integral F n dS directly without using Gauss' Divergence Theorem. (b) Evaluate the triple integral Ш div(F) dV directly without using Gauss' Diver- gence Theorem Note: You should obtain the same answer in (a) and...
a. Sketch the solid S:= {[x; y; z] in |R3 | x,y,z ≥ 0, and 2x...
a. Sketch the solid S:= {[x; y; z] in |R3 | x,y,z ≥ 0, and 2x + 4 y + 2z ≤ 12}. b. Using your calculator evaluate, i) as a triple integral and ii) by the divergence theorem, the volume of S. c. Find i)the surface area of the solid S and ii)the flux thru the top of S due to the vector field F, where F(x,y,z) = ( x + yz , y + xz , z +...
r 37. Singular radial field Consider the radial field (x, y, z) F (x2 + y2...
r 37. Singular radial field Consider the radial field (x, y, z) F (x2 + y2 + z2)1/2" a. Evaluate a surface integral to show that SsFonds = 4ta?, where S is the surface of a sphere of radius a centered at the origin. b. Note that the first partial derivatives of the components of F are undefined at the origin, so the Divergence Theorem does not apply directly. Nevertheless, the flux across the sphere as computed in part (a)...
(1 point) Let F(2, y, z) be a vector field, and let S be a closed...
(1 point) Let F(2, y, z) be a vector field, and let S be a closed surface. Also, let D be the region inside S. Which of the following describe the Divergence Theorem in words? Select all that apply. L A. The outward flux of F(x, y, z) across S equals the triple integral of the divergence of F(2, y, z) on D. IB. The outward flux of F(x, y, z) across S equals the surface integral of the divergence...
Evaluate the surface integral lis(r,y,z) (x, y, z) ds where f(x, y, z) = x +...
Evaluate the surface integral lis(r,y,z) (x, y, z) ds where f(x, y, z) = x + y + z and o is the is the surface of the cube defined by the inequalities 0 < x < 5,0 Sy < 5 and 0 <3 < 5. [Hint: integrate over each face separately.] 1 f(x, y, z) ds =
Evaluate the line integral f F dr for the vector field F(x, y, z) curve C...
Evaluate the line integral f F dr for the vector field F(x, y, z) curve C parametrised by Vf (x, y, z) along the with tE [0, 2 r() -(Vt sin(2πt), t cos (2πi), ?) ,
Evaluate the line integral f F dr for the vector field F(x, y, z) curve C parametrised by Vf (x, y, z) along the with tE [0, 2 r() -(Vt sin(2πt), t cos (2πi), ?) ,
Consider the vector field F(x, y, z) -(z,2x, 3y) and the surface z- /9 - x2 -y2 (an upper hemisph...
Consider the vector field F(x, y, z) -(z,2x, 3y) and the surface z- /9 - x2 -y2 (an upper hemisphere of radius 3). (a) Compute the flux of the curl of F across the surface (with upward pointing unit normal vector N). That is, compute actually do the surface integral here. V x F dS. Note: I want you to b) Use Stokes' theorem to compute the integral from part (a) as a circulation integral (c) Use Green's theorem (ie...
5. Setup (but do not evaluate) one integral (of any type) to find the flux of vector field F through surface S, where S s the unit cube given by 0 < x < 1,0 < y 1.0 < z 1, 5. Set...
5. Setup (but do not evaluate) one integral (of any type) to find the flux of vector field F through surface S, where S s the unit cube given by 0 < x < 1,0 < y 1.0 < z 1,
5. Setup (but do not evaluate) one integral (of any type) to find the flux of vector field F through surface S, where S s the unit cube given by 0
first picture is the notation on the test paper
Second picture is the question i would like to solve
thanks~
Notation and convention: r x +y The distance from the origin to the point r [x,y,z] + ê: The unit vector along the direction of r-[x, y,z] (a.e,6)-i.j.):m :The orthonormal bases of a Cartesian coordinate system. for dummy indices Einstein convention: Omitting the summation notation (repeated indices). Examples:ab,-a b, ab a b Notice: No dummy index is allowed to be...
5. A surface is parameterized by u and v as xacosucosv y bsinucosv z csinv Here 0su<2r, -/2<v< /2, a, b and c are nonzero constants. (a) Express the surface equation in the form f(x, y,z) 1. Find f(x,y,2) (b) Calculate the gradient of the f mentioned above and find the unit normal vector n ar ar (c) Calculate the (un-normalized) normal vector N=- X Cu xr, and normalize it to get the unit normal vector n again. Notation and...
(1) Let P denote the solid bounded by the surface of the hemisphere z -Vl-r-y? and the cone2y2 and let n denote an outwardly directed unit normal vector. Define the vector field F(x, y, z) = yi + zVJ + 21k. (a) Evaluate the surface integral F n dS directly without using Gauss' Divergence Theorem. (b) Evaluate the triple integral Ш div(F) dV directly without using Gauss' Diver- gence Theorem Note: You should obtain the same answer in (a) and...
r 37. Singular radial field Consider the radial field (x, y, z) F (x2 + y2 + z2)1/2" a. Evaluate a surface integral to show that SsFonds = 4ta?, where S is the surface of a sphere of radius a centered at the origin. b. Note that the first partial derivatives of the components of F are undefined at the origin, so the Divergence Theorem does not apply directly. Nevertheless, the flux across the sphere as computed in part (a)...
(1 point) Let F(2, y, z) be a vector field, and let S be a closed surface. Also, let D be the region inside S. Which of the following describe the Divergence Theorem in words? Select all that apply. L A. The outward flux of F(x, y, z) across S equals the triple integral of the divergence of F(2, y, z) on D. IB. The outward flux of F(x, y, z) across S equals the surface integral of the divergence...
Evaluate the surface integral lis(r,y,z) (x, y, z) ds where f(x, y, z) = x + y + z and o is the is the surface of the cube defined by the inequalities 0 < x < 5,0 Sy < 5 and 0 <3 < 5. [Hint: integrate over each face separately.] 1 f(x, y, z) ds =
Evaluate the line integral f F dr for the vector field F(x, y, z) curve C parametrised by Vf (x, y, z) along the with tE [0, 2 r() -(Vt sin(2πt), t cos (2πi), ?) ,
Evaluate the line integral f F dr for the vector field F(x, y, z) curve C parametrised by Vf (x, y, z) along the with tE [0, 2 r() -(Vt sin(2πt), t cos (2πi), ?) ,
Consider the vector field F(x, y, z) -(z,2x, 3y) and the surface z- /9 - x2 -y2 (an upper hemisphere of radius 3). (a) Compute the flux of the curl of F across the surface (with upward pointing unit normal vector N). That is, compute actually do the surface integral here. V x F dS. Note: I want you to b) Use Stokes' theorem to compute the integral from part (a) as a circulation integral (c) Use Green's theorem (ie...
5. Setup (but do not evaluate) one integral (of any type) to find the flux of vector field F through surface S, where S s the unit cube given by 0 < x < 1,0 < y 1.0 < z 1,
5. Setup (but do not evaluate) one integral (of any type) to find the flux of vector field F through surface S, where S s the unit cube given by 0
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