17. If f(x,y,z) and g(x, y, z) are continuously differentiable scalar functions defined over the oriented surface S with boundary curve C, prove that (Vf X Vg)•n do = O f Vg•dr.

Q: Show the function f(z) = z (z) is conformal at 0 and f'(0) = 0. Does this function violate our…

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Q: Suppose that z = xy² and x and y are differentiable functions of t with x = 1, y = −1, dx/dt = −2,…

A: Given  z=xy2 Also at t=-1 x=1y=-1dxdt=-2dydt=3

Q: then show that u and v satisfy the Cauchy-Riemann equation everywhere for ( i) and for all z # 0 in…

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Q: Use Green's Theorem to evaluate the line integral |v - x)dx + (2x – y)dy for the given path. C: x =…

A: Answer

Q: 6. In each case, compute the flux (x°, y°, z³). W- (a) F(x, y, z) = and a? + y? + 2² = 9 anc…

A: Use divergence theorem

Q: If F= 2y I- 3J +x²k and s the surface parabolic cylinder y= 8x in bounded by the planes OF the 2.…

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Q: Consider 6x (1+x²)² 2(x+y) (1+x²)² + 2y ý = Using the function V(x, y) = 12 + y² show that the…

A: What is Lyapunov Stability Theorem: Lyapunov stability theorem says that if a function is positive…

Q: Suppose the closed curve C in R? starts at the point (-1,1), goes along the parabola y = x2 to 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: 28. Show that the circulation of F(x, y, z) = (x², y², z(x² + y²)) around any curve C on the surface…

A: Given figure Stoke's Theorem Here take single integral over closed path C and double integral over…

Q: 6. Evaluate the line integral g. x²y² dx + xy dy where C consists of the arc of the parabola y = x²…

A: (a) Direct method The given line integral is ∮C x2 y2 dx + xy dy. The curve C consists of    C1 :…

Q: 10. Evaluate the integral cz(z-1) where C is a simple closed curve in the positive direction that…

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Q: Use Green's Theorem to evaluate the line integral |(y - x)dx + (2x – y)dy for the given path. C: x =…

A: Let's find.

Q: 8.(6 pts) Use Fundamental Theorem of Line Integrals to compute / Vf - dr where f(z, y, z) = ry? +…

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Q: It is easy to see that u(z) = Im (+) is harmonic in the unit disk |2| <1 and lim,→1- u(reio) = 0 for…

A: Given, Uz=Im1+z1-z2,z&lt;1 Now,…

Q: 12.2.7. Suppose that Q := {(x, y) e R : x > 0 and y > 0} and that f is a continuous function on R?…

A: Sol

Q: [39] Consider the heat conduction in a circular cylinder: x² + y² < a²,0 < z <b. Assume that the…

A: Given Data: Heat conduction in a circular cylinder  x2+y2&lt;a2, 0 &lt;z&lt;b To Find: (a) Boundary…

Q: Evaluate the line integral f. (2x – 4xy)dx + 3xy – 2x³y)dy over the straight path L joining point…

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Q: 1. If F= (y,z,x), the outward flux through a smooth closed surface (a) depends on the surface. (b)…

A: Given, F=y,z,x We know,  Flux by the Gauss divergence theorem. ∬F.ds=∬∫divF.dvF=yi+zj+xk If you…

Q: a) Evaluate the line integralZ F dr C in terms of t where F = cosyi+xj+ye²>k and C is the line…

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Q: Find the equation of the tangent plane to the surface z = ln (2x + y) at the point (-1,3,0) .…

A: To find the equation of the tangent plane to the surface z=ln2x+y at the point -1, 3, 0.

Q: Let F = (-z2, 2zx, 4y - x2}, and let C be a simple closed curve in the plane x + y + z = 4 that…

A: Given F={-z2,2zx,4y-x2}. Curve in the plane (s) = x+y+z=4. Area enclosed is 16.

Q: Find all point on the plane x+ yt z= 5 in the first octant at which f(x, y,z) xy'z' has a maximum…

A: We have to find all points on the plane x+y+z=5 in the first octant at which fx,y,z=xy2z2 has a…

Q: Show that the function f(z) = e-z",z # 0 and f(0)=0 is not analytic at z=0, although Cauchy-Riemann…

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Q: Use Green's Theorem to evaluate the line integral (y – x)dx + (2x – y)dy for the given path. C: x =…

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Q: 1. Evaluate the line integral f(z, y) ds where f(x, y) = Vx+ 2y and C is the straight-line segment…

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Q: Suppose that (z) is entire and that the harmonic function u(x, y) = Re[fz)] has an upper bound uo;…

A: As requested following is the answer for question 12.

Q: Use Green's Theorem to evaluate the line integral √(x - (y - x)dx + (2x - y)dy for the given path.…

A: Let's find.

Q: A surface x(y, z) is defined implicitly xy + yz +x'z' = 3-xZ Calculate ду

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Q: Suppose that z is a differentiable function of x and y with dz (4, 7) = 4. ду dz (4, 7) = 60 and- ax…

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Q: |Evaluate the line integral: f.(r² + y) dx + (4x – cos(y)) dy where C is the boundary of the square…

A: According to Greens theorem, the line integral ∮CPdx+Qdy=∬DQx-PydA. BY the extended version of…

Q: Use the Green's Theorem to evaluate the line integral 2xydx + x² – y² dy where C is the curve…

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Q: A necessary condition for the functional defined by X2 = dx f(x, Y(x),Y'(x)) to have a local minimum…

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Q: Find the surfaces on which the function u = 2 (x2 + y²)/z is constar

A: Since u is a constant so, x2+y2z=4a , where a is an arbitrary constant.

Q: Suppose we have a surface f (x,y)=x²y+cos (4x) · Which of the following is the gradient of f(x,y) at…

A: First I have found the partial derivatives of f(x,y)  with respect to 'x' and 'y' at point P(π/2,…

Q: (3) +. R Ris bounded by the semicircle y = V2x – x² and the line y = x

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Q: Suppose that z is a differentiable function of x and y with dz (1, 1) = 162 and(1, 1) = 6. dz ду dx…

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Q: Please work out the results of the following differential operations • Gradient of f (x, y, z) = x²…

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Q: 4. Evaluate ||| x y dV , where E is bounded by the parabolic cylinders y = x², y' = x and the planes…

A: The region E is bounded by the parabolic cylinders y = x2, x = y2 and the planes z = 0 and z = x +…

Q: Let C denote the curve of intersection between x = -5z° and y = 6z", starting from A(0,0,0) and…

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Q: Suppose that z is a differentiable function of x and y with dz (6, 1) = 35 and(6, 1) = 7. dy dz Ifx…

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Q: Use Green's theorem to evaluate the closed line integral: . x²y dx +x'y dy along C between -1 < x< 2…

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Q: 3. Show that the line integral | (r + y)dx + xdy is independent of path and evaluate this integral…

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Q: Consider the following. Z = =DVx²+ y?, 5x - 2y + 3z = 22, (3, 4, 5) (a) Find symmetric equations of…

A: a) Here, f(x,y,z)≡x2+y2-z2=0g(x,y,z)≡5x-2y+3z=22P≡(3,4,5)

Q: Consider a potential problem in the half-space defined by z 0, with Dirichlet boundary conditions on…

A: The green function solution to the potential problem with this boundary for Dirichlet boundary…

Q: 3.4.6 Evaluate the line integral I = =f( (xy dx + x dy) C where C is the closed curve consisting of…

A: We have to solve the given problem

Q: 3.4.6 Evaluate the line integral [sin y dx + (x- cos y) dy] taken in the anticlockwise sense, where…

A: Green's Theorem States that  Let C be the positively oriented, smooth, and simple closed curve in a…

Q: 12. Find the outward flux of F = x*yi-cyj+zk across the boundary of the region which is bounded…

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Q: *6. Let f: R' →R be given by =+4x'z-6xyz+4y-3x'y. Is M=f-((0}) a smooth surface (2-dimensional…

A: Given, fxyz=z2+4x3z-6xyz+4y3-3x2y2 Now, writing the given function in matrix form.…

Q: Let F(r, y, z) = (y² + 2z, 2ry + 4, 2a), and let the curve C be described by x = t, y = t, z = t3…

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