Vector Analysis Homework: Finding the Unit Normal Vector at P

[manimaran1605]

Homework Statement

This picture is taken from Div, curl, grad and all that by schey, while finding the unit normal vector of the surface S (defined as w=f(x,y)) at a point P, to find the normal vector he considered a two tangent non-collinear vectors (u and v) at a point P, to find u he considered a plane passing through point P parallel to xz plane, the plane which intersects the surface S traces a curve C which contains the point P, he drawn a tangent to the curve at the point p, let the x component be ux, My question is how the z-component of u is (∂f/∂x)ux ?

Homework Equations

No equations found

The Attempt at a Solution

I have an idea that z-component of u is some approximation, but i havn't learn multivariable calculas a lot, so please enlighten me. Thank you

 

[CAF123]

You are correct. In a small displacement from P, the linear approximation is a good one and you can write $z(x= P + u_x) \approx z(x=P) + \frac{\partial z}{\partial x} u_x$ (The Taylor expansion truncated, ignoring higher order terms). For an infintesimal displacement, this approximation becomes exact.