Surface integrals: Calculating the flux over a particular surface is a very common way of determine the electric field. Eyaluate the integral f, v dA where v(r, y, z) = 3zr â +5x ý+2y 2 and S is the rectangular surface lying in x-z plane from (0,0,0) to (2,0,3). Choose the direction of +ŷ to be indicative of positive flux.

Q: 1. Let F(x, y) = (-y,x). Let C be the boundary of the triangle with vertices (0,0), (1,0), (0,1),…

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Q: A magnetic field with an intensity Fo cos (2wt) (F, and w are positive constants) is applied to an…

A: Here, we are given the following system of wave equation with boundary conditions.…

Q: (yx + 3xsiny + 3y) dx + (x° cosy – y° + 3x) dy along the curve C that is a square oriented…

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Q: 1. Let F(x, y) = (-y,x). Let C be the boundary of the triangle with vertices (0,0), (1,0), (0,1),…

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Q: The electric field at a point P(z, y, z) due to a point charge q located at the origin is given by:…

A: Given  E=q r∥r∥3  where  r=x i∧+y j∧+zka div  E=q∂∂xi∧+∂∂y j∧+∂∂z        xx2+y2+z23/2…

Q: QI Given the electric flux density, D= xya, + xya, + xya, C/m2. By using divergent theorem find Q…

A: NOTE: We are entitled to solve only the first question, unless specified. Formula Used:Divergent…

Q: (0,4) C R (-2,0) (-1,0) (1,3) (c) Use the WASHER METHOD to set up a (sum of) definite integral(s)…

A: Divides the region into two subregions.

Q: Green’s Theorem for line integrals Use either form of Green’sTheorem to evaluate the following line…

A: By Green's Theorem, we have where R is the region enclosed by curve C.

Q: 2. Let F = −y³i + x³j + cos z³k and C be the curve generated by the intersection of the cylinder x²…

A: a. Given, F¯=-y3i¯+x3j¯+cos z3 k¯ Let C be the curve generated by the intersection of the cylinder…

Q: 7. Find the circulation of the field F(x, y) = xi+yj around the circle r(t) = (cos t)i + (sin t)j.

A: Since you've asked multiple questions, as per our answering guidelines we are supposed to answer…

Q: Green’s Theorem for line integrals Use either form of Green’sTheorem to evaluate the following line…

A: By Green's theorem, we have where R is the region enclosed by curve C.

Q: Using Green's Theorem, find the outward flux of F across the closed curve C. F = (-5x+ 2y) i + (8x -…

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Q: A Express the volume of the solid lying under the surface f(z, y) = In(a(x² + y²) + 6) and above the…

A: Given: The function f(x,y)=lna(x2+y2)+6   and the disk x2+y2

Q: 13-16 Evaluate the double integral over the rectangular re- gion R. fa 16. /| (x sin y- y sin x) dA;…

A: The given double integral is: ∬Rx siny-y sinx dA The rectangular region is given by: R=x,y: 0≤x≤π2,…

Q: A- Use Greens theorem to evaluate the circulation or flux for the velocity 1 y) = x+y j across or…

A: The flux form of Green’s theorem relates a double integral over region D to the flux across boundary…

Q: F(x, y) = (6yey — 9)i + 6xej (a) Show that F is a conservative vector field. fy - 9x = 9m (b) Find a…

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Q: exp[-2(x2 + y²)] and over the region in the plane z = 0 defined Let W be the three dimensional…

A: Given : 1) W is a three dimensional region under the graph of fx, y=e-2x2+y2 and over the region in…

Q: 4-x² 4. Evaluate [*, [* (r² + y°) dz dy dx. 4-. Hint: This iterated integral is a triple integral…

A: Cylindrical Coordinates:       Put x= r cosθ and y=r sinθ and z=zthen,      r = x2+y2 and θ =…

Q: 35. Flow integrals Find the flow of the velocity field F = (x + y)i – (x² + y²)j along each of the…

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Q: 40. The flux line integral of F = (e*-, e'-x), where C is the boundary of { (x, y): 0 < y s x,0 < x<…

A: Given: C is the boundary of .

Q: 2 V2y-y a Evaluate the integral V² + y²dzdydx by using cylindirical coordinates. 0 0 0

A: Solve by converting into cylindrical coordinates (r,theta,z)

Q: The area is bounded by the curve y = ex, y = e-xand x = 1. (units in meters)  Find the area, in m2.…

A: 1) Given:         curve  y=ex                    y=e-x and x=1   The point of intersection of two…

Q: Let F-[4z?, x2, 2y²] and C be the positively oriented boundary of the triangle with vertices (0, 0,…

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Q: 1. Let F = 2ry?7+ (2yz² + 2y) , and let Ğ = (y+ 2)i+ (y-a)j, where Ci is the curve consisting of the…

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Q: 3- Use Stokes theorem to compute the line integral SF · dr, where F = [e-3y, e², e-2*] and C is the…

A: Given that F=e-3y,ez,e-2x The objective is to find ∫CF.dr. Where C is z=2x2,0≤x≤2,0≤y≤1.

Q: F. dr as a line integral and as a double integral. C F(x, y, z) = (-y + z)i + (x − 2)j + (x - y)k S:…

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Q: Evaluate the line integral using Green's Theorem and check the answer by evaluating it directly. $ 6…

A: Given problem:-

Q: G. Let Glx,y)=(y+cosx)i+(2r-siny)j. Find the line integral of G'around the perimeter of the square…

A: Given information :  G→ = (y + cosx)i + (2x - siny)j Square with vertices at (0,0), (3,4), (-1,7)…

Q: r(u, v) = (2 cos u, 2 sin u, v), 0 sus 2n, 0 < v < 3 is a parametrization of the surface S given by…

A: Given,

Q: 65. Transform the surface integral curl(F dS into a line integral using Stokes' theorem, and…

A: We’ll answer the first question since the exact one wasn’t specified. Please submit a new question…

Q: PUse Green's Theorem to evaluate the line integral. S(x² – y²) dx + 2xy dy C C: boundary of the…

A: We have to evaluate the integral ∫Cx2-y2dx+2xydy by using Green's theorem. Where C is the boundary…

Q: 13.3 - Fundamental Theorem for Line Integrals 1. An object of mass M is located at the origin,…

A: Solution...

Q: Use a parameterization to find the flux The flux is SSF n da of F = 2²1+xj-32k in the outward…

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Q: Using Stokes's theorem, calculate the line integral is the triangle with vertices (0,0,0), (1,0,0),…

A: To use Stokes's theorem to calculate the line integral of the below function. F=2y2, 3x2, z−x The C…

Q: S ={(x, y, z) E R³ |x = y² – z², 0 < y < 1, 0 < z < 1}. (a) Compute the surface integral over S of…

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Q: Find the flux of the vector field F across the surface S in the indicated direction. F = zk, S is…

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Q: (3) Evaluate the flux integral where S is the surface given by 22 =1 +y°, 0<= s 4, oriented upwards…

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Q: (x², y², z(x² + y²)) around any closed curve on the surface of Show that the circulation of F = the…

A: Circulation of vector field F around a close curve C = ∫CF.ds If parametric equation of curve C…

Q: Evaluate the surface integral I. -4x dS. S Where S is the triangular region with vertices (4, 0, 0),…

A: Surface integral of function f(x,y,z) on surface S given by z=g(x,y)…

Q: .2 Evaluate the line integral xy dx + x-dy, where C is the path going counterclockwise around the…

A: Given, ∮Cxydx+x2dy, Where C is the path going counterclockwise around the boundary of the rectangle…

Q: (3) Evaluate the flux integral where S is the surface given by 22 =r + y², 0< z< 4, oriented upwards…

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Q: Find the net outward flux: if the solid D is the solid right circular cylinder, 0 < z <1 and x² + y²…

A: The given solid D is as follows: The boundary surface S can be divided in three different parts…

Q: Use the surface integral in Stokes' Theorem to calculate the flux of the curl of the field F = 2zi +…

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Q: Find the surface integral of the field F(x,y,z) = - i+5 j+2 k across the rectangular surface z= 0,…

A: Given query is to find surface integral inthe direction of k.

Q: se the Divergence Theorem to evaluate       S   F · N dS      and find the outward flux of F…

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Q: please only answer part (b) thank you

A: For any vector field F bounded by a curve C with positive orientation and S is the smooth surface.…

Q: Evaluate the line integral ∫c F.dr , where F=[xy,yz,xz] and C is the boundary of the hemisphere x2…

A: Given problem is : Evaluate the line integral ∫c F.dr , where F=[xy,yz,xz] and C is the boundary…

Q: oply Green's theorem to evaluate the line integral fxydx - xydy, where C is the boundary of the…

A: See the attachment

Q: Using Green's theorem in the plane, evaluate $ (ry² dy - x²y* dx). where C is the triangle with…

A: Using Green's theorem in the plane, we have to evaluate ∮Cx3y2dy-x2y3dx, where C is the triangle…