I have written the following code to determine the equation of a plane tangent to a surface at a given point:
z[x_, y_] := Log[x y^2 - 2]
zx[x_, y_] := Evaluate@D[z[x, y], x]
zy[x_, y_] := Evaluate@D[z[x, y], y]
tanPlane[x_, y_, x0_, y0_] :=
Expand[z[x0, y0] + zx[x0, y0] (x - x0) + zy[x0, y0] (y - y0)]
tanPlane[x, y, 3, 1]
It works well. But I'm now interested in writing code to find the equation of a tangent plane to a given parametric surface at a specific point.
So, for example, say $x = u + v, y = 3u^2, z = u-v$; at point (2,3,0).
How would I write code to calculate the eq. of the tangent plane to this surface? I've got this so far, which gives me the normal vector.
x[t] := u^2
y[t] := v^2
z[t] := u + 2 v
R[t] := {x[t], y[t], z[t]}
Cross[D[R[t], u], D[R[t], v]]
What I want to do is to auto-evaluate the given point, which will solve for the right $u$ and $v$ and then write and simplify the eq. of the plane.