2. Solve Laplace's equation inside the circular annulus given in the polar coor- dinates by the inequalities a a > 0. Consider the boundary conditions: u(a, 0) = 0, and u(b, 0) = g(0).

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: Which one of the following is the parametric equa- tions of the line that is normal to the surface…

A:

Q: Solve the following boundary value problems for the Laplace equation on a rectangle of sides and 2.…

A:

Q: Match the surface to its corresponding equation in spherical coordinates. |p = 4 cos() 1.5 -1 -0.5 0…

A: To match the given surface to their given Equation in spherical coordinates.

Q: 3.4.8 Evaluate the area of the surface z = 2-x² - y² lying above the (x, y) plane. (Hint: Use polar…

A:

Q: QUESTION 6 Use the Green's Theorem to evaluate the line integral xfydx +xy dy. where C is the…

A: Using the given vertices we can draw the required triangle and then apply Green's theorem to…

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

A:

Q: 2. Let R(t) = sin 4t î – bcos"(21) j + 2t k. Find the moving %3D trihedral of the curve of R(t) at…

A: Given

Q: 3.1 Use the Cauchy integral formulas to evaluate the eiz cos z dz, Le 2(2z – 37) where C is the…

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

Q: For a particle with position 7(1) = ()i+ (51) j + (?² – 16r) k, when is its speed at a minimum?

A:

Q: Which one of the following is the parametric equa- tions of the line that is normal to the surface…

A:

Q: 15. Find the radius of curvature at 0 = -n of the parametric equations x = 3(0 + cos 0), y = 3(1 –…

A: For the parametric curve x=x(θ), y=y(θ)Curvature K=|x'y''-y'x''|[(x')2+(y')2]32Radius of…

Q: 1. Let C be a path along the upper half of a circle from the point P(2,0) to the point Q(-2,0).…

A:

Q: The parametric form of the solutions of the PDES Uy + u u z = 0 , u(x,0) = -x is

A:

Q: In three dimensions, the cylinder x? + z2 – 10x – 6z – 2 = 0 has radius k i +tj+ and the axis line…

A: Radius =6  r(t)= (5+6cos(thetha))i+tj+(3+6sin(thetha)k

Q: Evaluate cos z dz , where C is the circle |z – 1| = 3. (Integrate counterclockwise around C.)

A:

Q: 1.3. Derive the mass conservation statement for spherical coordinates [eq. (1.10)] by writing eq.…

A:

Q: In changing the differentiating variable to a polar coordinate system, A. dx anddy becomes dr and…

A:

Q: 3.4.4 By transforming to polar coordinates, show that the double integral (x² + y²) ² dx dy (xy)²…

A:

Q: Example 14. Use polar coordinates to evaluate the double integral 16-z2 dy dx (9 + x2 + y²)³/2 ° I

A:

Q: Problem 52. Use Stoke's theorem to evaluate fy°dx – x³dy + z³dz where C is the trace of the cylinder…

A: Given integral and curve are,

Q: B.2. Express to its standard form, determine the center, radius ar 10. x² + y? + 10x – 14y + 58 = 0…

A: # We are entitled to solve one question at a time , please resubmit the other question if you wish…

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: Evaluate the unoriented line integral (2æ² + y² + z²) ds along the parametric curve a : [0, 1] → R³…

A:

Q: 45. If A = (y – 2x )1 + (3x + 2y)j, compute the circulation of A about a circle C in the xy plane…

A:

Q: 29. Find parametric equations for the tangent line to the cur ve of intersection of the paraboloid z…

A: Given paraboloid is z=x2+y2 ellipsoid is x2+4y2+z2=9 and the point (1,-1,2) find the parametric…

Q: Solve the BVP for the disk (exterior Dirchlet problem) v²u = 0, 1< r <∞ u (1,0) = sin 0 + cos 20

A:

Q: 2. Solve Laplace's equation inside the rectangle 0 <r< 2, 0 <y <1, with the following boundary…

A:

Q: 5. Discuss the bifurcations of the system ŕ = r(µ – sin r), 0 = 1 as µ varies.

A: Given that: This system has no fixed points, but r can take on zero values where μ≤0. Where equality…

Q: When transforming the triple integral "(x² + y² + z²)³ dzdydx 2n n 3 to spherical coordinates T…

A: Its false

Q: 1.Let 3 le the solid obtained by rotating by region shown in the fiqure about the Assume as3 and b=…

A: Given: y=ax(x-b)2 and y=0 and  0≤x≤b The rotation about the y-axis. Assume a=3 and b=1

Q: QUESTION 11 A three-dimensional solid obtained by revolving a region in the xy-plane about a line is…

A: We have to check

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

A:

Q: 2. Let R(t) = }sin 4t î – bcos“(21) j + 2t Ř. Find the moving %3D trihedral of the curve of R(t) at…

A:

Q: Consider the Laplace Equation with 0≤x≤2, 0≤ y ≤1 and the meshsize equal to 0.5. The boundary…

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Q: Solve Laplace's equation inside a rectangle 0srS L, 0SYS H, with the following boundary conditions:…

A: Consider the solution as ux,y=XxYy. Then, the Laplace equation becomes Yy∂2Xx∂x2+Xx∂2Yy∂y2=0. Divide…

Q: Calculate the line integral f(-102 + 4z + 5) de + (-7z – Oy? – by – 4) dy, where Dis the annulus…

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Q: Let (r, 0) represent polar coordinates. Find a radial solution u(r) of the Poisson equation 1 3²u 1…

A: To get the solution to  First, we will solve for…

Q: 1. Draw a sketch of the cylinder z= ==V36 – 4y2.

A: Given: z=1336-4y2

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: When transforming the triple integral 9-x2 9-x2-y2 (x²+ y² + z²)% dzdydx 2n n 3 to spherical…

A: The formula for the conversion of triple cartesian to triple polar coordinate is…

Q: 4. (i) Use the parametric equations of an ellipse, z = a cos (t) , y = bsin (t), 0 < t< 2r, to find…

A:

Q: Consider the polar curves C2 :r = 4 cos 0 and C3 : r = 4 sin 0. (b) Set up the integral that gives…

A: The given curves are C2: r=4cosθ and C3: r=4sinθ .........(1) To find: The area under the given…

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: 1 ³u + r2 d02 1 ди dr2 r dr 0 =

A:

Q: Solve the BVP for the disk (exterior Dirchlet problem) v'u = 0, 1< r <o∞ u (1,0) = sin 0 + cos 20

A: The given function is ∇2u=0 The given condition is u(1,θ)=sinθ+cos 2θ We have to find the general…