# Vector analysis

1. May 24, 2014

### manimaran1605

1. The problem statement, all variables and given/known data

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 ?

2. Relevant equations
No equations found

3. 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

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2. May 24, 2014

### 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.

Last edited: May 24, 2014