Dynamic Optics in Economics: Quantitative, Experimental and Econometric Analyses 1st Edition and published by Peter Lang GmbH, Internationaler Verlag der Wissenschaften. The Digital and eTextbook ISBNs for Dynamic Optics in Economics: Quantitative, Experimental and Econometric Analyses are 9783631842546, 3631842546 and the print ISBNs are 9783631831915, 3631831919. Save up to 80% versus print by going digital with VitalSource. Additional ISBNs for this eTextbook include 9783631842553, 9783631842560.

Normal vector of aplane

For a 2D curve [latex]y=f(x)[/latex], there is at most one tangent line to a point [latex](x_0, y_0)[/latex] on the curve.  The equation of tangent line to 2D curve [latex]y=f(x)[/latex] at point [latex](x_0, y_0)[/latex] is

Tangentplaneand normal line

Now, you should engage with the 3D plot below to understand the tangent plane[1]. Follow the steps below to apply changes to the plot and observe the effects:

3D Interactive Plots for Multivariate Calculus Copyright © 2022 by Dr. Na Yu, Ryerson University is licensed under a Creative Commons Attribution-NonCommercial-ShareAlike 4.0 International License, except where otherwise noted.

The tangent plane in 3D is an extension of the above tangent line in 2D. For a 3D surface [latex]z=f(x,y)[/latex], there are infinitely many tangent lines to a point [latex](x_0, y_0, z_0)[/latex] on the surface; these tangent lines lie in the same plane and they form the tangent plane at that point.

tangent plane中文

Recall that two lines determine a plane in 3D space. Thus, one usually uses two special tangent lines to determine a tangent plane and these two tangent lines are related to the partial derivatives (i.e., [latex]f_x[/latex] and [latex]f_y[/latex]) of the surface function [latex]z = f(x,y)[/latex]. The equation of the tangent plane to surface [latex]z = f(x,y)[/latex] at point [latex](x_0, y_0, z_0)[/latex] is