How Do You Calculate Gaussian Curvature of a Plane in Mathematica?

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Homework Statement

Use Mathematica to calculate the Gaussian curvature of the plane Ax+By+Cz=D, in which A, B, C, and D are constants and C≠0.

Use the following data:

Homework Equations

The Attempt at a Solution

First I found the line curvature. As here:

Simplify[ArcCurvature[{v1*t + x, v2*t + y, v3*t + z}, t]]

That code gets the result = 0.
So, I know that the curvature needs to be 0, even with the Gaussian curvature.

I have looked at Gaussian curve code in Mathematica but I can't quite figure out how to get it to work.
I was told that I need to translate it to a 'u' and 'v' parameterized surface, but I have no idea how to do that...especially in Mathematica.
Here is my Gaussian Curvature attempt:

GaussianCurvature[{v1*t + x, v2*t + y, v3*t + z}, t]

I just can't seem to get it to be 0. What am I doing wrong?

Thanks in advance

 

[mighty2000]

for any help or guidance you can provide.
Thank you for bringing this question to our attention. I would be happy to assist you in finding the Gaussian curvature of the given plane in Mathematica.

Firstly, let me clarify that the Gaussian curvature of a plane is always 0, as you have correctly observed. This is because a plane is a flat surface and does not have any curvature.

Now, to find the Gaussian curvature of a parameterized surface in Mathematica, we need to first define the surface in terms of the u and v parameters. Let's say we have a surface given by the following equation:

Ax + By + Cz = D

To parameterize this surface, we can use the following equations:

x = u
y = v
z = (D - Au - Bv)/C

Now, using these equations, we can define our surface in Mathematica as follows:

surface[u_, v_] := {u, v, (D - A*u - B*v)/C}

Next, we need to find the first and second fundamental forms of this surface, which are required to calculate the Gaussian curvature. The first fundamental form is given by:

E = (surface_u). (surface_u)
F = (surface_u). (surface_v)
G = (surface_v). (surface_v)

Similarly, the second fundamental form is given by:

L = - (surface_uu). (surface_n)
M = - (surface_uv). (surface_n)
N = - (surface_vv). (surface_n)

Note: The surface_n here represents the normal vector to the surface, which can be calculated using the cross product of surface_u and surface_v.

Now, we can use the following formula to calculate the Gaussian curvature:

K = (LN - M^2) / (EG - F^2)

Plugging in the values of E, F, G, L, M, and N in the above formula, we can find the Gaussian curvature for our given plane.

Here is the complete code in Mathematica:

surface[u_, v_] := {u, v, (D - A*u - B*v)/C}
surface_u = D[{surface[u, v]}, u]
surface_v = D[{surface[u, v]}, v]
surface_uu = D[{surface[u, v]}, {u, 2}]
surface_uv = D[{surface[u, v]}, u, v