1. In the plane, let D be a simply-connected region whose boundary ðD is a piecewise C', simple, closed curve, oriented counterclockwise. Let în be the outward unit normal vector to D. Given two functions, f(x, y) and g(x, y), both C² on an open set containing the domain D, show that: (a) · ds = - for any piece-wise C1 closed curve C in the domain D. (b) I. v9) dzdy = f. (s9) ·î ds – where Vg = V. ỹg= grx + gyy. (c) ·î ds

Q: Let S be the portion of the cylinder y = ln x in the first octant whose projection parallel to the…

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Q: Find the normal line for the level surface f(x,y,z) = 3x3 + cos (2y) + ln (2z) = c at the point…

A: f(x,y,z)=3x3+cos(2y)+ln(2z)-c=0Fx=9x2 , Fy=-sin(2y) , Fz=12z×2=1z∇F =<Fx,Fy,Fz >=<9x2,…

Q: 16. Let C be a simple closed smooth curve that lies in the plane x + y + z = 1. Show that the line…

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Q: Find the line integral with respect to arc length (8x + 9y)ds, where C is the line segment in the…

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Q: Find the tangent plane for the level surface f(x,y,z) = 5x4 + cos (4y) + In (6z) = c at the point…

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Q: Let f(z) = x² +iy² where x and y are the real and imaginary parts respectively of z, and C is a…

A: Given:                                                                fz=x2+iy2 C is a contour…

Q: Let S be the sphere x^(2)+y^(2)+z^(2)=4 with outward normal vector. Use the divergence theorem to…

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Q: For F = xy?a, + yz²a, – x²za,, use divergence theorem to evaluate ff, F. dS where S is the sphere of…

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Q: 18. Suppose that V ·F, = V • F, and V × F, = V × F, over a re- gion D enclosed by the oriented…

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Q: 21. Show that the volume V of a region D in space enclosed by the oriented surface S with outward…

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Q: Find the normal line for the level surface f(x,y,z) = 4x² + cos (3y) + In (5z) = c at the point…

A: The given surface is: f(x,y,z)=4x2+cos(3y)+ln(5z)

Q: Use Stokes' Theorem to caleulate ſ. F - dř, where F = zyi+3=3+5zk, and C is the closed path around a…

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Q: Find the normal line for the level surface f(x,y,z) = 5x3 + cos (5y) + In (6z) = c at the point…

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Q: 17) Find the area of te senface genereted by revolving the curve X=3¢, y= 2f about fue x-qxis on tue…

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Q: d) An open surface F is given by F = {(x,y, z) |z = 4 – x² – y² , z > 0} . Compute the flux of V…

A: V(x,y,z) = <2x+3y,2y+3x,-4z>F={ (x,y,z) : z=4-x2-y2 , z≥0}Consider, g(x,y,z) = z+x2+y2-4…

Q: Let C be the curve in space given by the following parameterization. r(t) = tỉ + t2j+ 9k, 0 <ts v7.…

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Q: b) Use Gauss theorem to evaluate F.ndS, where F(r, y, 2) = (2x²y² + In z) i – xy³ j – tanh(a² + y³)…

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Q: Find the tangent plane for the level surface f(x,y,z) = 3x5 + cos (5y) + In (5z) = c at the point…

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Q: 5. Consider the saddle surface r = y- 2, find the point where the tangent plane is parallel to x+y+z…

A: If two vectors a1, b1, c1 and a2, b2, c2 are parallel, then it can be written that a1, b1, c1=ka2,…

Q: Compute ,F . dS for F = (y, 5,-x), S being the portion of the plane x + y + z = 1 in the octant x,…

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Q: Let F = 1xi + 5yj and let n be the outward unit normal vector to the positively oriented circle x² +…

A: To find the flux we have to first compute unit normal vectors by using formulan=Gradient…

Q: Find the line integral with respect to arc length / Q = (0,3). (2x + 7y)ds, where C is the line…

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Q: Let G(x. y)=x*yi+(y+xy*)j. Evaluate the line integral [G-dr, where Cis the curve x= ť and y=t* for…

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Q: Let n be the outer unit normal of the elliptical shell S: 9x +25y +225z = 225, z20, and let %3D F=…

A: ∇×F=ijk∂∂x∂∂y∂∂zyx2x2+y432sin…

Q: Use Green's Theorem to calculate the work done by the force F on a particle that is moving…

A: Given ,                                                                             F(x,y) =…

Q: Find the line integral with respect to arc length ∫C(6x+4y)ds, where C is the line segment in the…

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Q: Compute / f ds over the curve specified. (1) f(x, y, z) = z², r(t) = for 0 < t < 2

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Q: Find a unit normal vector to the surface f(x, y, z) = 0 at the point P(4, – 5, – 5) for the fur 5x…

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Q: Let f(x, y) = x³ + 2ry – y² be a func the tangent plane to f(x, y) at the poi

A: Here we have f(x,y)=x3+2xy-y2  we have to find the equation of the tangent plane to f(x,y) at point…

Q: Find the tangent plane for the level surface f(x,y,z) = 7x³ + cos (4y) + In (4z) = c at the point…

A: Tangent plane for the level surfacefx,y,z at (a, b, c) is…

Q: Let S be a square plate with boundary C, having each side of length 2 units, lying in the plane x =…

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Q: Compute the scalar line integral xy ds where C is the part of the circle of radius 3, centered at…

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Q: Suppose F(x, y) = -yi + xj and C is the line segment from point P (3,0) to Q = (0, 2). (a) Find a…

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Q: 5. Show that the equation of the tangent plane to f(r, y, z) = c at (ro, Yo: 20) is given by the…

A: This is the question from vectors.

Q: Use Green's theorem to calculate the work done by force F on a particle that is moving…

A: GivenF(x,y)=x32-8yi+5x+8yj,C: Boundary of a triangle with vertices (0,0), (5,0) and (0,5)To…

Q: Evaluate || F:ñ dS where S is the surface bounded by z = a - Vx² + y² and the xy-plane including the…

A: The vector field is F→=xπ, yπ, 3z-2π. The surface is bounded by z=a-x2+y2 and xy-plane. The function…

Q: Let F = yi + xj + x³y³ z³k, C the curve x2 + y² + 6x – 2y + 9 = 0, z = 2 (with counterclockwise…

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Q: Q19// A scalar filed V = xyz exists over a surface S defined by x? + y? = 9 bounded by the plane…

A: Given scalar field, V=xyz This field exists over a surface S defined by x2+y2=9 bounded by the plane…

Q: Let F(r, y) = 4xî + 3yî and let i be the outward unit normal vector to the positively oriented…

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Q: Let F(x,y,z)=(2ye?,2e²,12(2-e?)). Let S be the portion of the graph z=1n(2x2+40y3+2) which is above…

A: Given that: Fx,y,z=2yez,2ez,122−ez Here S is the portion of the below graph which is above the given…

Q: Let F(x,y,z)= (4ye?,2e?,12(2-e?)). Let S be the portion of the graph z= In(4x2+60y3+2) which is…

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Q: Suppose C is a positively oriented, piecewise-smooth, simple closed curve. Verify that the result of…

A: Given    ∫Cxy2dx-x2ydy Where C is arc of parabola y=x2 from -1,1 to 1,1 and the line segment 1,1 to…

Q: Use the Divergence Theorem to find F- ndS, where F = (y°z,x²z,z') and the surface S is the sphere x…

A: See the details solution below

Q: 2. Suppose that S is a surface in R3 with unit normal vector N surface area of S is given by (n1,…

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Q: Q19// A scalar filed V = xyz exists over a surface S defined by x? + y? = 9 bounded by the plane…

A: Since you have posted multiple questions. I am answering the first one. Please post the question…

Q: Find the tangent plane for the level surface f(x,y,z) = 6x3 + cos (4y) + In (4z) = c at the point…

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Q: Use Divergence theorem to find the ouward flux of F = 2xz i - 3xy j – z2 k across the boundary of…

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Q: Find the normal line for the level surface f(x,y,z) = 7x5 + cos (5y) + In (9z) = c at the point…

A: The given level surface is: f(x,y,z)=7x5+cos(5y)+ln(9z)Given point is:  (9, 3, 10)

Q: Find the line integral with respect to arc length (2x + 6y)ds, where C is the line segment in the…

A: The solution of line integral is given as follows :