Functions of several variables - three dimensional space, Mathematics

Assignment Help:

Functions of Several Variables - Three Dimensional Space

In this part we want to go over a few of the basic ideas about functions of much more than one variable.

Very first, keep in mind that graphs of functions of two variables, z = f (x, y) are surfaces in three dimensional (3D) space. For instance here is the graph of z = 2x2 + 2y2 -4.

1143_Functions of Several Variables - Three Dimensional Space 1.png

This is an elliptic parabaloid and is an instance of a quadric surface. We saw so many of these in the earlier section. We will be seeing quadric surfaces quite regularly later.

Other common graph that we will be seeing quite a bit in this course is the graph of a plane.  We comprise a convention for graphing planes which will make them a slightly easier to graph and hopefully visualize.

Remind that the equation of a plane is illustrated by

ax + by + cz = d

or if we solve this equation for z we can write it in terms of function notation. This provides,

f (x, y) = Ax + By + D

To graph a plane we will usually find the intersection points along with the three axes and then graph the triangle which connects those three points. This triangle will be a part of the plane and will provide us a fairly decent thought on what the plane itself should act like.  For instance let's graph the plane illustrated by,

f (x, y) = 12 - 3x - 4 y

For the aim of graphing this it would possibly be easier to write this as,

 z = 12 - 3x - 4 y                                ⇒         3x + 4 y + z = 12

Here now, each of the intersection points along with the three main coordinate axes is described by the fact that two of the coordinates are zero.  Example for this, the intersection with the z-axis is illustrated by x = y = 0.  Thus, the three intersection points are,

x - axis : (4, 0, 0)

y - axis : (0, 3, 0)

z - axis : (0, 0,12)

Below is the graph of the plane.

69_Functions of Several Variables - Three Dimensional Space 2.png

Now here, to extend this out, graphs of functions of the type w = f (x, y, z) would be four dimensional surfaces.  Actually we cannot graph them, although it does not hurt to point this out. We next wish to talk about the domains of functions of much more than one variable.  Remind that domains of functions of a single variable, y = f (x), contained all the values of x that we could plug into the function and get back a real number. At present, if we think about it, the meaning of this is that the domain of a function of a single variable is an interval (or intervals) of values from the number line or one dimensional space.

The domain of functions of two variables that are, z = f (x, y), are regions from two dimensional space and contain all the coordinate pairs, (x, y) , that we could plug into the function and obtain back a real number.


Related Discussions:- Functions of several variables - three dimensional space

Fractions, how do you convert in a quicker way?

how do you convert in a quicker way?

Proper fractions, find all the kinds of fraction and give an 10 examples.

find all the kinds of fraction and give an 10 examples.

What is the sample space for this experiment, A die is thrown repeatedly un...

A die is thrown repeatedly until a six comes up. What is the sample space for this experiment? HINT ;A= {6} B={1,2,3,4,5,} Ans: The sample space is = {A, BA, BBA, BBBA, BBBBA.

Briefly explain markov chains, Question 1 An experiment succeeds twice as ...

Question 1 An experiment succeeds twice as often as it fails. Find the chance that in the next six trials there will be at least four successes Question 2 An insurance compan

Differential Equations, Verify Liouville''''s formula for y "-y" - y'''' + ...

Verify Liouville''''s formula for y "-y" - y'''' + y = 0 in (0, 1) ?

John 47 out of 86 free-throws who best free-throw shooter, Michael made 19 ...

Michael made 19 out of 30 free-throws this basketball season. Larry's freethrow average was 0.745 and Charles' was 0.81. John made 47 out of 86 free-throws. Who is the best free-th

Limits-of-sum, limit 0 to 2(3x^2+2) Solution) integrate 3x^2 to x^3 and...

limit 0 to 2(3x^2+2) Solution) integrate 3x^2 to x^3 and 2 to 2x and apply the limit from 0 to 2 answer is 12.

Two consecutive positive integers whose product is 90, What is the lesser o...

What is the lesser of two consecutive positive integers whose product is 90? Let x = the lesser integer and let x + 1 = the greater integer. Because product is a key word for m

Write Your Message!

Captcha
Free Assignment Quote

Assured A++ Grade

Get guaranteed satisfaction & time on delivery in every assignment order you paid with us! We ensure premium quality solution document along with free turntin report!

All rights reserved! Copyrights ©2019-2020 ExpertsMind IT Educational Pvt Ltd