I'm having some trouble with a problem from my Calculus IV course. The problem states:
The textbook I've been assigned for my calculus courses has proved time and time again to be utter garbage, and so there are literally zero examples of this sort of problem in my book. The best I get is these two sentence:
The figure included shows the equation as ϕ=ϕ0 and the graph is an upside-down cone with the point at the origin. Obviously, I know that the equation I was given is a cone, so that takes care of the first part of describing the graph. I also know that the graph of an elliptical cone in rectangular coordinates is given by z2=a2x2+b2y2, so it would follow that the top half of an elliptical cone would produce an upside-down cone with the point at the origin. I'm then looking for an equation of the form z=a2x2+b2y2
So now I'm left utterly confused as to how I might find the specific values of a and b that correspond to a given phi of 3pi/4. And once I do that, I'm even more lost as to what the equation might be in cylindrical coordinates. I don't even know how to make a cone in that coordinate system. Any help would be greatly appreciated.
In Exercises 29-38, describe the graphs of the equations in R3, and provide alternative equations in the specified coordinates systems.
38) Change ϕ=43π to the rectangular and cylindrical coordinate systems.
The textbook I've been assigned for my calculus courses has proved time and time again to be utter garbage, and so there are literally zero examples of this sort of problem in my book. The best I get is these two sentence:
In the figures that follow, we show that a sphere, a plane, and a cone. These are the graphs obtained [in spherical coordinates] when ρ,θandϕ, respectively, are constant.
The figure included shows the equation as ϕ=ϕ0 and the graph is an upside-down cone with the point at the origin. Obviously, I know that the equation I was given is a cone, so that takes care of the first part of describing the graph. I also know that the graph of an elliptical cone in rectangular coordinates is given by z2=a2x2+b2y2, so it would follow that the top half of an elliptical cone would produce an upside-down cone with the point at the origin. I'm then looking for an equation of the form z=a2x2+b2y2
So now I'm left utterly confused as to how I might find the specific values of a and b that correspond to a given phi of 3pi/4. And once I do that, I'm even more lost as to what the equation might be in cylindrical coordinates. I don't even know how to make a cone in that coordinate system. Any help would be greatly appreciated.