# Chapter 7 - Chapter Chapter 7 Functions of Several...

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hapter 7 Chapter 7 Functions of Several Variables

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Overview troduction Introduction Functions of 2 variables Domain of 2 variables Geometric Representation Partial Derivatives Geometric Interpretation Higher Order Partial Derivatives
Overview he Chain Rule The Chain Rule irectional Deri ati es Directional Derivatives Geometric Meaning h i l M i Physical Meaning Functions of Three Variables

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Overview aximum and Minimum Values Maximum and Minimum Values Local Maximum and Minimum ritical Points Critical Points
Introduction

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Introduction bjective: Objective: To extend some methods of single-variable differential calculus to functions of several variables. In many practical situations, the value of a quantity may depend on more than one variable.
Introduction - Example h r V 2 r h Output of a factory ------ amount of capital invested and the size of manpower. Current in electrical circuit ------ capacitance, electromotive force, impedance and resistance in the circuit. p

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Functions of 2 Variables ( , ) ------ a rule that assigns to each ( , ) a real number ( , ), where and are real. f xy fx y x y ( , ) ------ is a function of and z y z x y and independent variables dependent variables z 12 In general, ( , , , ) ------- function of variables n zf x x x n
Domain of 2 Variables of Domain D f f defined} is ) , ( | ) , ( { y x f y x

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Domain of 2 Variables - Example 25 Let ( , ) 3 sin . f x y x y x y  {( , ) | , are real} f D xy xy
Domain of 2 Variables - Example 22 Let ( , ) 1. fx y x y  Positive 10 xy  {( , ) | 1} f Dx y x y y x 0

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Domain of 2 Variables - Example 1 Let ( , ) . fx y xy {( ) | 0 and 0} x y x y {( , )| 0 and 0} f Dx  y x 0
eometric Geometric Representation

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Geometric Representation ( ) ------ a curve in -plane yf x x y ( , ) a surface in 3-D space zf x y
Geometric Representation - Example 1 z xy  1 xyz  z 1 y x z 1 1 1 0 y x artesian Equation of plane: 00 0 Cartesian Equation of plane: , where . ax by cz d d ax by cz 

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Geometric Representation - Example ( , ) ------ a surface in 3-D space z fx y Pause and Think !!! Question: How to "plot" the surface 2 2 y x z
2 2 2 . radius with ) 0 , 0 ( center circle r r y x y r x -r r -r

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Geometric Representation - Example ( , ) ------ a surface in 3-D space z fx y Pause and Think !!! Question: How to "plot" the surface 2 2 y x z 22 1 x y Fix 1 z ircles Fix 2 z 2 x y Circles
Geometric Representation - Example 2 2 y x z z 2 2 22 Circle xyc y x z 0 y For a fix value of z, we get a circle x Any plane parallel to the plane intersects the surface ------ a circle xy

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2 x parabola yx y x
Geometric Representation - Example ( , ) ------ a surface in 3-D space zf x y Pause and Think !!! Question: How to "plot" the surface 2 2 y x z 2 z x Fix 0 y arabola Fix 1 y 2 1 zx Parabola

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Geometric Representation - Example 2 2 y x z z arabola 2 2 2 Parabola z xc y x z 0 y For a fix value of y, we get a parabola x Any plane parallel to plane intersects the surface ------ a parabola xz
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Chapter 7 - Chapter Chapter 7 Functions of Several...

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