Parabolic coordinate system question

I figured it out.In summary, the homework question involves finding the volume of a 3D solid bounded by two paraboloids. The binding condition is given in cartesian coordinates as -1+(x2+y2) < 2z < 1-(x2+y2). The task is to rewrite this condition in parabolic coordinates and use the already derived metric tensor to solve for the volume. The attempt at a solution involves finding the functions of s, t, and p and determining the limits of integration. After some trial and error, the individual was able to figure out the solution on their own.
  • #1
Felicity
27
0

Homework Statement



a 3D solid is bounded by 2 paraboloids. The binding condition in cartesian coordinates is

-1+(x2+y2) < 2z < 1-(x2+y2)

a) rewrite the binding condition in parabolic coordinates
b) using parabolic coordinates and the (already derived) metric tensor, find the volume of the solid

Homework Equations



x=stcos(p) y= stsin(p) z= (t2-s2)/2

The Attempt at a Solution




I found the binding conditions to be equal to

-1 + s2t2 < t2 - s2 < 1 - s2t2

I have the metric tensor and I know i just need to do a triple integral and multiply by the square root of the metric tensor but how do I find the functions of s, t and p and how do I know the limits of integration?

I've tried splitting it into two inequalities and moving the variables around looking for a pattern but I can't really see anything.


any help would be greatly appreciated

thank you

-Felicity
 
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  • #2
Never mind
 

FAQ: Parabolic coordinate system question

What is a parabolic coordinate system?

A parabolic coordinate system is a coordinate system in which a point in 3D space is represented by three coordinates: a radial coordinate, an azimuthal coordinate, and a parabolic height coordinate. It is often used to describe the motion of objects in a gravitational field.

How is a parabolic coordinate system different from a Cartesian coordinate system?

In a Cartesian coordinate system, a point in 3D space is represented by three coordinates: x, y, and z. In a parabolic coordinate system, the coordinates are r, φ, and z. The r coordinate represents the distance from a fixed point (the focus), the φ coordinate represents the angle from a fixed reference line (the axis), and the z coordinate represents the height above a fixed plane (the directrix).

What are some applications of the parabolic coordinate system?

The parabolic coordinate system is commonly used in physics and engineering to describe the motion of objects in a gravitational field, such as satellites orbiting a planet or projectiles launched from the Earth's surface. It is also used in fluid dynamics to describe the flow of fluids around a parabolic surface, such as an airfoil.

Can the parabolic coordinate system be extended to higher dimensions?

Yes, the parabolic coordinate system can be extended to higher dimensions. In 4D space, it is represented by four coordinates: r, φ, θ, and ψ. In general, a parabolic coordinate system in n-dimensional space is represented by n coordinates: r, φ1, φ2, ..., φn-2, and z.

How is the parabolic coordinate system related to other coordinate systems?

The parabolic coordinate system is closely related to other coordinate systems, such as cylindrical and spherical coordinate systems. In fact, the parabolic coordinate system can be seen as a combination of these two systems, with the addition of a parabolic height coordinate. It is also related to the elliptic coordinate system, which is similar but uses an elliptical surface instead of a parabolic one.

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