Show that the quantity (r0)2 = [ (ct)2 - (x2+y2+z2)] is unchanged by the Lorentz transformation.
Show that the quantity (r0)2 = [ (ct)2 - (x2+y2+z2)] is unchanged by the Lorentz transformation.
The gravitational field F(x,y,z) =cx /(x2 + y2 + z2)3/2 e1+ cy /(x2 + y2 + z2)3/2 e2+ cz/ (x2 + y2 + z2)3/2 e3 is a gradient field, where c is a constant, such that the field is rotation free. If we define f(x,y,z) = −c /(x2 + y2 + z2)1/2 , then show that (a) F = grad(f). (b) curl(F) = 0.
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= 1 and Find the volume inside both x2 + y2 + z2 x2 + y2 = x.
the intersection of X2= Y2 + Z2 and the plane Y= 2
2. (20 pts) Find the surface area of that part of the sphere x2 + y2 + z2-4 that lies inside the paraboloid z x2 + y2.
2. (20 pts) Find the surface area of that part of the sphere x2 + y2 + z2-4 that lies inside the paraboloid z x2 + y2.
Consider the solid enclosed by x2 + y2 + z2 = 2z and z2 = 3(x2 + y2) in the 1st octant. a) Set up a triple integral using spherical coordinates that can be used to find the volume of the solid. Clearly indicate how you get the limits on each integral used. b) Using technology, or otherwise, evaluate the triple integral to find the volume of the solid.
2. Find the area of the part of the surface of the sphere x2 + y2 + Z2-42 that lies within the paraboloid z-x2 + y2. Hints: * Complete the square for ×2 + y2 + Z2-42+ (it is a sphere with center (0, 0,) Find the intersection to determine the region of integration
2. Find the area of the part of the surface of the sphere x2 + y2 + Z2-42 that lies within the paraboloid z-x2 + y2....
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your steps.
4. Conpute the volume of the region s inside the cylinder z2 +y2 = 1, between the paraboloid :-x2 + y2-2 and the plane z + :-4
4. Conpute the volume of the region s inside the cylinder z2 +y2 = 1, between the paraboloid :-x2 + y2-2 and the plane z + :-4
(a) Find and identify the traces of the quadric surface x2 + y2 ? z2 = 25 given the plane. x = k Find the trace. Identify the trace. y=k Find the trace. Identify the trace. z=k Find the trace Identify the trace. (b) If we change the equation in part (a) to x2 ? y2 + z2 = 25, how is the graph affected? (c) What if we change the equation in part (a) to x2 + y2 +...
Write the equation in spherical coordinates. (a) x2 + y2 + Z2 = 64 (b) x2 - y2 - 2 = 1 | eʼsin?(p)cos? (0) – eʼsin(@)sin?(0) – e cos?(p) = 1
mmer 2019 3. Evaluate: M y2 dV where E the solid hemisphere x2 + y2 +z2 9 and y 2 0 indrnse)
mmer 2019 3. Evaluate: M y2 dV where E the solid hemisphere x2 + y2 +z2 9 and y 2 0 indrnse)