MultiVariable Calculus - partial derivatives

Question

“Given the tangent plane of an implicit surface, written in the form f_x(x - x0) + f_y(y - y0) + f_z(z - z0) = 0, how do we find the line tangent to the level curve z = z0?”

Answer 1

First off, tangent plane is generally written as F_x(x-x_o) + F_y(y-y_o) + F_z(z-z_o) = 0 in standard textbooks. This is because upper case F and lower case f have a different meanings in Calculus.

To solve this problem, first we need to find the gradient of the 3D solid, which is the vector ∇f(x, y) =〈F_x, F_y〉. Recall that a gradient is always orthogonal to the level curve. From this gradient vector ∇f(x, y), we can easily construct another vector that is orthogonal to it, and let's name it as v = 〈a, b〉. We then use this new vector v as the guiding vector for our tangent line to the level curve as in this equation in parametric form:

x = x_o + at

y = y_o + bt

z = z_o.

I hope this solution helps. Thank you.