4) Solve the Laplace's equation inside a cylinder of radius R and length L with the given boundary data. Au = 0 u(r, 0, 0) = 0 u(r, 0, L) = R² – 2 u(R, 0, z) = 0
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- Answer the following question please.Use cylindrical coordinates to evaluate X d V R when Ri x'+y?What is the solution to the boundary value problem for the Laplace equation in two dimensions, subject to the boundary conditions given by a circle of radius "R" centered at the origin?Show that the general solution to the two-dimensional Laplace's Equation. aiv B = 0 is (Acosh(rx) +Bsinh(tx))(Ccos(tx) +Dsin(tx)), where t2 is a ax? separation constant. A,B,C, and D are constants (which are unique, given boundary conditions). Note, we covered the 1-D Laplace's equation in ECE3250. V: 3 Ver.y) aThe parametrization r(0,0) = ((3+2 cos 0) cos , (3+2 cos 0) sind, 2 sin 0) describes a surface that is shaped like a donut (it's also called torus). i. Draw the 0 = 0 grid curve, that is, sketch the curve given by r(0, 0) in R³. ii. In the same plot, draw the = π/2 grid curve, that is, sketch the curve given by r(0, π/2) in R³.2. Solve Laplace's equation inside the rectangle 0Please help with the solution. This is vector calculus. I am having issues with the limits of integration for y.2 2 A circular wire hoop of constant density 8=2 lies along the circle x² + y² = 8a² in the xy-plane. Find the hoop's moment of inertia, I₂, about the z-axis.1. Given that p(x, y, z) = 2x² + y² + z² find Vo. Deduce the magnitude and the direction of the greatest rate of change of at the point (1, 1, 1). Further, calculate the outward pointing unit normal to the ellipsoid 2x² + y² +2²= 4, at the point (1, 1, 1). Use this to find the Cartesian equation of the tangent plane at this point.Recommended textbooks for youAdvanced Engineering MathematicsAdvanced MathISBN:9780470458365Author:Erwin KreyszigPublisher:Wiley, John & Sons, Incorporated
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