Suppose a point in three-dimensional Cartesian space, (X, Y, Z) , is equally likely to fall anywhere on the surface of...
suppose a point in three-dimensional Cartesian space. (X, Y, Z), is equally likely to fall anywhere on the surface of the hemisphere defined by X2 + Y2-22-1 and Z20. (a) Find the PDF of Z, /zz) (b) Find the joint PDF of X and Y, /x. ylx, y). suppose a point in three-dimensional Cartesian space. (X, Y, Z), is equally likely to fall anywhere on the surface of the hemisphere defined by X2 + Y2-22-1 and Z20. (a) Find the...
roblem 4 points A point A (X, Y, Z) in a three-dimensional Euclidean space R3 has the uniform joint distribution within the ball of radius 1 centered at the origin (OinR3.) Consider a random variable, T d (A, O), that is the distance from A to the origin. 1. Find the cumulative distribution function for T 2. Evaluate its expectation, E T] 3. Evaluate the variance, Var [T] .
5. [12 Marks) Consider the level surface of the function f(x, y, z) defined by f(x, y, z) = x2 + y2 + x2 = 2a?, (1) where a is a fixed real positive constant, and the point u = (0,a,a) on the surface f(x, y, z) = 2a. a) Find the gradient of f(x, y, z) at the point u. b) Calculate the normal derivative of f(x, y, 2) at u. c) Find the equation of the tangent plane...
6. (12pts) Use the divergence theorem to find the flux F.ndS with outward pointing normal n with F(x, y, z) =< x2,-y, z >, where s is the surface of the hemisphere z = V 1-x2-y2 and its base in the xy plane. 6. (12pts) Use the divergence theorem to find the flux F.ndS with outward pointing normal n with F(x, y, z) =, where s is the surface of the hemisphere z = V 1-x2-y2 and its base in...
For the elliptic-paraboloid find the Cartesian equation of the tangent plane at the point on the surface where x = -2 and y = 1. Your answer should be an equation, expressed in terms of the Cartesian variables x, y and z using the correct syntax. For example: 3*x-2*y+5*z=2, or, 2*(x-1)+4*(y-2)+z-1=0, or 3*x+ 6*z=12-y, or y-x+35*(z-256)=20 Do not use decimal approximations all numbers should be entered as exact expressions, for example 5/2 2x2+y2-10-2, 0 < z < 10 2x2+y2-10-2, 0
The temperature at any point (x, y, z) in space is T = x y3 z4 Find the highest temperature on the surface 4 x2 + 4 y2 + z2 = 8. Enter the exact value of your answer in the box below. Warning: If your answer involves a square root, use either sqrt or power 1/2. Note: The highest temperature cannot be 0.
a) A vector field F is called incompressible if div F = 0. Show that a vector field of the form F = <f(y,z),g(x,z),h(x,y)> is incompressible. b) Suppose that S is a closed surface (a boundary of a solid in three dimensional space) and that F is an incompressible vector field. Show that the flux of F through S is 0. c)Show that if f and g are defined on R3 and C is a closed curve in R3 then...
13. Evaluate Is F.dš that is, find the flux of the field across the surface. F(x,y,z)=-4z ī + y) – 3x K , S is the hemisphere z = 14 – x2 - y2; ñ points upward.
Let F(x, y, z) = 4i – 3j + 5k and S be the surface defined by z = x2 + y2 and x2 + y2 < 4. Evaluate SJ, F.nds, where n is the upward unit normal vector.
Let F(x,y,z) = 4i – 3j + 5k and S be the surface defined by z= x2 + y2 and 22 + y2 < 4. Evaluate SJ, F. nds, where n is the upward unit normal vector.