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6 fox, y, z)= / r = distance between 2 particules Po (to, Yo, 2o) P...
Extra Credit Prove that the V fo at each point Po (xo. yo, zo) on the surface f(x(t),y(t),z(r)) = K for some constant K is orthogonal to the tangent vector T() of each curve C described by the vector function on the surface passing through Po (xo,yo, zo). Hint, remember that the tangent vector T(o) R'(), so prove that Vfo R'O) 0
Extra Credit Prove that the V fo at each point Po (xo. yo, zo) on the surface f(x(t),y(t),z(r))...
5. Calculate the surface area of the portion of the sphere x2+y2+12-4 between the planes z-1 and z ะไ 6. Evaluate (xyz) dS, where S is the portion of the plane 2x+2y+z-2 that lies in the first octant. 7. Evaluate F. ds. a) F = yli + xzj-k through the cone z = VF+ア0s z 4 with normal pointing away from the z-axis. b) F-yi+xj+ek where S is the portion of the cylinder+y9, 0szs3, 0s r and O s y...
2/6 2" (10%) (a) Sketch the graphs of the two functions y=z(4-z) and y=z and mark the finite region (R) enclosed between them. (Identify (R) carefully!) (b) Let W be the volume of the solid of revolution obtained by revolving the above region (R) around the y-axis. (i) Use the shell method to write down the integral for W. (No need to evaluate the integral.) (ii) Repeat part(i) by using the disk method.
2/6 2" (10%) (a) Sketch the graphs...
Notation and convention: r x +y The distance from the origin to the point r [x,y,z] + ê: The unit vector along the direction of r-[x, y,z] (a.e,6)-i.j.):m :The orthonormal bases of a Cartesian coordinate system. for dummy indices Einstein convention: Omitting the summation notation (repeated indices). Examples:ab,-a b, ab a b Notice: No dummy index is allowed to be repeated more than twice. You should change the "names" of the dummy indices before taking the product of two summations...
Suppose E is the half-cylinder described by 2y2 between z and the ry-plane where y 2 0. Suppose further that the density at each point in E is proportional to the distance from the z-axis. (a) (6 points) Find an expression for the mass of E as a triple integral. Then briefly explain why this integral is difficult to compute Suppose E is the half-cylinder described by r' +ґ-1 between z = 4 and the xy-plane where y > 0....
Suppose E is the half-cylinder described by 2y2 between z and the ry-plane where y 2 0. Suppose further that the density at each point in E is proportional to the distance from the z-axis. (a) (6 points) Find an expression for the mass of E as a triple integral. Then briefly explain why this integral is difficult to compute
Suppose E is the half-cylinder described by 2y2 between z and the ry-plane where y 2 0. Suppose further that...
PLEASE DO LETTER d.)
PLEASE DO LETTER f.)
The plane from e.) is 4(x-2)+6(y-1)+(z-1)=0 or 4x+6y+z=15
15. The temperature on an unevenly heated metal plate positioned in the first quadrant of the xy-plane is given by 25xy + 25 C(x, y) = 7 (x - 1)2 + (y - 1)2 +1° Assume that temperature is measured in degrees Celsius and that x and y are each measured in inches. (Note: At no point in the following questions should you expand...
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draw graph. Also label the answers thoroughly. Answer both for
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m2 6 Fig awre . 2. The magnitude of the force of attraction between the positively charged proton and the negatively charged electron in a Hydrogen aton is given by f(r) = ke2/r2 where e is the charge on the proton, k is Coulomb's constant, and r is the distance between the proton and the electron. Assume that the protorn can't move. The...