What Are the Different Coordinate Systems Used in Calculus?

[JasonJo]

1) Rewrite the following expression in polar coordinates:
(second derivative of z with respect to x) + (second derivative of z with respect to y)

where x=rcos(theta)
y = rsin(theta)

i had first derivative of z with respect to x = (dz/dx)(dx/dr) + (dz/dx)(dx/d(theta))

same concept for y

and then i just took the derivative of my dz/dz and dz/dy variables again

2) describe this surface in cylindrical coordinates
z = rcos3(phi)

express in cartesian coordinates

i really do not get cartesian coordinates

can anyone show me the generic method?

3) x = psin(phi)cos(theta)
y = psin(phi)sin(theta)
z = pcos(phi)

near which points of R^3 can we solve for: p phi and theta in terms of x, y and z, describe the geometry behind your answer.

again, i am really really weak in coordinate systems.

any help is appreciated, thanks guys

 

[mathmike]

what does z = in #1

x = p sin phi cos theta

y =p sin phi sin theta

z = p cos phi