515P) Calculate (r is the usual position vector): 7:7, 7(r-), r •7(r) integrate this over a...
Calculate (r is the usual position vector: (cose/r?), 7(rr), Yx(FxB) where B is constant vector along z direction.
515P) Calculate (r is the usual position vector: (coser), (ri), 7 x(x B) where B is constant vector along z direction.
7. Calculate the divergence over the volume of a sphere of radius 3 in a vector field where =4rsin- cos.(r, with Deduce the flux through the surface of the sphere. 7. Calculate the divergence over the volume of a sphere of radius 3 in a vector field where =4rsin- cos.(r, with Deduce the flux through the surface of the sphere.
How do I prove that the vector line integral of F over the curve C not depend on the value of R? R, 9) = - 2 + y2 x2 + y2 and CR is the circle of radius R centered at the origin. 7
PROBLEM4 (a) Write an expression for the mass density ρ(r) of a point particle with a mass m at a position r; How much is the volume integral of ρ over the entire space? (b) What is the mass density of a system of two point particles with mass m each. One of the particle is at the origin of the coordinate system and the other one is at ä. How much is the integral of the density over the...
13. (5 points) Reverse the order of integration for the following iterated integral. You do not have to integrate. cos y dy dx 14. (5 points) Integrate the function g(r,0) = p sin over the sector of a disc in the first quadrant bounded by the circle r² + y2 = 1, the circle r² + y2 = 4, the line y = rV3, and the r-axis. 15. (5 points) Convert the following iterated integral from Cartesian to polar. You...
i) Consider the following position vector: r = x + y + z2. Convert this position vector into spherical coordinate system.
r 37. Singular radial field Consider the radial field (x, y, z) F (x2 + y2 + z2)1/2" a. Evaluate a surface integral to show that SsFonds = 4ta?, where S is the surface of a sphere of radius a centered at the origin. b. Note that the first partial derivatives of the components of F are undefined at the origin, so the Divergence Theorem does not apply directly. Nevertheless, the flux across the sphere as computed in part (a)...
Integrate A=(sinx, yx, cosz) over the outer surface of a cube with side length 1. For simplicity, place one vertex of the cube at the origin and align its three neighboring edges along the positive x, y, z axis, respectively. For each face of the cube, calculate A•n before doing the integration, where n is the norm of that face (the unit vector perpendicular to the surface that points outward).
Assume that a Spherical Planet Of Radius R, Has a Uniform Mass Density (Per Unit Volume) Distribution Throughout, Of Value Po. Also, Assume that There Is a Massive Dust Cloud In the Rest Of the Universe, Which Decays Exponentially In Radius, r, Away From the Surface Of the Planet, Where the Mass Density Varies As ρ(r) = Po exp| | | |, For r2R- a) Using the Integral Form Of Gauss's 6. Law, [n.gda--4πGJsoh', And Spherical Coordinates (Specifically Using the...