(a) Given a surface function, z(x, y) == + xy with x cos0 and y= sin 0, 2 %3D az and əz (i) find ax az (ii) find by using chain rule.

Q: 1. Use Green's theorem in order to compute the line integral $ (3cos + 6y?) dx + (sin(5") + 16x³) dy…

A: Here the given integral is                               ∮C3cosx+6y2dx + sin5y+16x3dy

Q: 1.5 (3-y Evaluate the integral dx dy using polar coordinate 'y

A: Given

Q: QUESTION 8 Find the area of the cylindrical surface in Xyz-space where a point (X,y,z) on the…

A: Find the surface area of the cylinder in xyz-space where a point x, y, z on the surface satisfies…

Q: Example 3 a) Evaluate the surface integral 2x²y dS over the surface y² + z² = 1 between x = -1 and x…

A: Note: We are entitled to solve only one question at a time. As the specified one is not mentioned,…

Q: 18. Show that the centroid of the sector in Figure 13 has y-coordinate 2R sin o (0, y) R FIGURE 13

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Q: 1. a) Use the cylindrical coordinates to evaluate (3 + xy)dV, where E is the solid E in the first…

A: Here is the solution of the given problem.

Q: (a) Find the solution of the limit using polar coordinate system. x² + y2 lim (x,y)-(0,0) x2 + y?…

A: Since you have asked multiple question, we will solve the first question for you. If you want any…

Q: Assuming that C is the counterclockwise oriented boundary curve of the unit square in the first…

A: It is an simple and easy question on vector calculus  Can be easily solve by use of green's theorem

Q: 16 The area under the curve 4y? (x2 + 1)3 was rotated about a line located to the left 9 of y – axis…

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Q: Evaluate f (x + 2y)dx + (y – 2x)dy around the ellipse C defined by x = 4 cos 0, y = %3D | C 3 sin…

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Q: The solution of Laplace's equation Urx + Uyry 0 < r < 3, 0 < y< 4, subject to the boundary…

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Q: The surface area of the surface generated by revolving y= 1 – x² for 0<x<1 about the x – axis is s=…

A: The solution is as follows

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: 3. 3) 6) X= y 〇s454 6) Find the area 0f the surface of rotation Created by ocyEd curve about the 3.…

A: Since you have asked multiple question, we will solve the first question for you. If you want any…

Q: In two dimensions, the motion of a fluid (an incompressible and irrotational fluid) is governed by a…

A: The velocity potential                                      φ(x,y)=9sinπxsin6πy then we have to find…

Q: The dynamic equilibrium of a one-story building is described by the following equation: x(t) = xc(t)…

A:

Q: 2. Show that the surface area of the solid of revolution obtained by revolving the curve y = 1/x, x…

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Q: The surface area of the surface generated by revolving y=1-x² for 0<x<1 about the x- axis is s=…

A: If you like the solution then please give it a thumbs up.... The Answer is:              S =…

Q: Given (1 + y³)y' = x². (i) Determine a region of the xy-plane for which the given differentia…

A: Explanation of the solution is given below...

Q: (a) Suppose that a black hole has magnetic field F = (2a* + e" sin(2), y³ + xe*, 2z³ + e" cos(z) +…

A: TO FIND: Magnetic flux across the black hole

Q: .... ........ 4.8. Find the area of the surface generated by revolving the curve x = cost, y =…

A: See the details solution in below

Q: 1. Set up the formulation to find the surface generated when the curve y = x² is revolved about the…

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Q: 5. On the surface z-3x^2+2y^2 (an elliptic paraboloid), (x,y) changes from (1,1) to (0.99,1.02). Use…

A: Since you have asked multiple questions, we will solve the first question (Question 5) for you. If…

Q: The shape of a redwood tree (without any branches) can be modeled by an upside down paraboloid, f(x,…

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Q: 5) Evaluate the line integral S YX²dX + XY²dY on curve C, where C is the upper portion of the www m…

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Q: 9-x2 C: Change to polar and evaluate the double integrals S (x3 + xy?)dydx V9-x2

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Q: The surface area of the solid of revolution generated by rotating x = (y² + 2)²,1 < y s 2 about the…

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Q: Consider the system i = rx + ax2 – x3 where a E R. When a = 0, we have the normal form for the…

A: Given system is x.=rx+ax2-x3 it can be written as x.=x(r+ax-x2) For the fixed point…

Q: 27) If the solid of revolution formed by rotating R2 about the y-axis is a cylindrical shell, then…

A: You have posted multiple questions. As per the instruction, we will be answering only the first one…

Q: y = cos(x), 0 <x<

A: Consider the given surface whose area is required to calculate.

Q: Problem 5 Compute the solid in the first octant bounded above by the surface z 9xy/1 – x² /4 – y?…

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Q: need to find the limits of integration in terms of ρ, φ, and θ.

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Q: Solve the boundary value problem for the heat equation: Ut = Ura, 00 u(0, t) = 0, u(1, t) = 0,…

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Q: Use numerical integration with Ax = 1.0 m to locate the centroid of the surface generated by…

A: Given That : y=3x216 and ∆x=1.0 m To find : locate the centroid of the surface generated by…

Q: A 3 cm length silver bar with a constant cross section area 1 cm (density 10 g/cm³, thermal…

A: The equation is u t…

Q: 3. Solve the Laplace equation inside a 60° wedge of radius 1 subjected to the boundary conditions…

A: The Laplace equation is given by: r2∂2u∂r2+r∂u∂r+∂2u∂θ2=0 The boundary conditions are: ∂u∂θr,0=0,…

Q: (a) Calculate the centroid (7, y) of the region under the graph y = cos x for 0 < x <T/2.

A: Hi, since you have asked multiple questions, we will solve the first question for you. If want any…

Q: (b) Find the surface area of the portion of the paraboloid z-2-x -y above the xy- plane. AMINATION A…

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Q: Find the exact area of the surface obtained by rotating the curve about the x-axis. y = cos -х 0< x…

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Q: Find the area of the surface generated when the curve y=1/4(e2x + e−2x)is rotated about the x-axis…

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Q: 1 9. Evaluate dV;where E is the solid between the spheres of radius 1 (x² + y² + z²)³/2 E and 2…

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Q: Find a cubic approximation of r(θ) = (θ 2 , eθ , θ) at θ = 0.

A: given rθ=θ2,eθ,θ to find - cubic approximation at θ=0

Q: A 3 cm length silver bar with a constant cross section area 1 cm" (density 10 g/cm, thermal…

A:

Q: 8. Evaluate the line integral (x'dx+ xe"dy where C is the closed path along the triangle in the…

A: Here we have to evaluate the integral ∫cx2dx+xe-ydy where C is the closed path along the triangle in…

Q: 1. Evaluate the line integral I = = [v¢4x° + y*) dx + x(2x° + 3y*) dy] around the ellipse x/a? +…

A: The given line integral is, I=∮y4x2+y2dx+x2x2+3y2dy We know the Green's theorem:…

Q: The volume of the solid generated by the curve y = e¯ª, between x = 2 and x = 7 about the x – axis…

A: Given curve, y=e-x x=2  and   x=7

Q: The volume of the solid generated by the curve y = e¬ª, between x = 3 and x = 7 about the |3| x –…

A: