Two Or More Variables Functions.docx

“Two or more variables functions:” There are many ways to visualize a two variables function, and this is the tridimensional way: Definition: The graph of a function f: D c R2 = R is the group of the (X; Y; Z) points which Z={(x,y,f(x,y)}|(x,y) € R, that is to say. Notice: The graph is a two variables function Z= f (X, Y) can be interpreted geometrically as a “S” Surface in the space so that. Its projection in the XY dimension is D, the f dominance. As a consequence, for each (X, Y) point in D it has an (X, Y, Z) point in the Y surface, otherwise for each (X, Y, Z) point there is a (X, Y) point in D.

First Example: Draw the function n F (X, Y) = x2 + y2 +1 Solution: The graph of this function is very common and are better known as paraboloids.

As it can be noticeable in the image, there are slices in a default height so that there is a difference when it comes to draw the groups over the graph. Similarly, it can be applied over the isothermal maps which shows off the temperature. Off course it can vary out, that is why the conventions exists.

Level curves A curve is a directionless line that changes it constantly.

A surface is body’s limit with the exterior. First, they’re not functions, they’re groups over the graph that look like slice of the same one, that is to say, the whole figure is not taken to represent it from one only perspective, but to slice it in one transversal axis so that the whole slice in the picture is drawn. It is well known that this shapes are tridimensional, and this is the main reason of why they must be drawn in three vectors; to represent this shapes in two dimensions, the grouped couples are used to draw them, it means “X” and “Y”, which are the actual dominium of the function, that is to say, the numbers that exist inside the graph of the function, but when the sliced figure is represented, just some numbers are taken into account. While “Z” express counter domain, that is to say, the space’s surface that fills the three dimensions. To write a function down: Z=”F(X;Y) of X;Y” Applications:

Notice that the last paraboloid z= x2 + y2 +1 has its symmetry axis in a parallel way to the “Z” axis, that is to say, that a paraboloid such as Y=X2 + Y2 +1 gets to have its symmetry axis over the “Y” one. Example2: Draw the functions F (X; Y) = -Z+Y Solution Process: This is one of the graphs that are going to be really frequently, it is about the Y+Z=2 graph:

Surfaces: because of being many of the S surfaces that are going to be used and do not come from Z= F (X; Y) it is necessary to extend the graph definition: Surface’s Definition: The graph of the function F (X, Y, Z) =0 is the set of points (X, Y, Z) E R3 that satisfies this equation, and usually it refers to the graph of an equation such as the S surface. Third Example: Show off that the sphere’s drawn line, over he graphs X + Y = 4 is an ellipse To figure the equation out the next system must be solved. X2 + Y2 + Z2 = 10 Y+Z=4 Z= 4 – 4} X2 + Y2 (4-4)2 = 1 Which turns out to be an ellipse: (x2 /2) + (y-2)2 = 1, Z= 4-4 It is not used to write down a curve like the last way because is hard to handle, it turns out much more comfortable and advantageous to work with flat curves. 𝑥 = √2 cos 𝜃 𝑦 = sin 𝜃 + 2 𝑧 = 2 − 𝑠𝑒𝑛𝜃 con ɸ E [2; π]

“Grupo editorial Iberoamericano. Curvas y superficies de nivel (3 y 4) saved from Http:1368750003_45507784-4-3-curvas y superficies de nivel”

Most used or known curves Circumference with its center in (H; J) and ratio r:

Ellipse with its center in (H; J) and semi axis in a and b: [(x-h)2 / a2 + [(y-j)2 /b2 = 1

The one with the hyperbole is: (x2)/a2 – (y2)/b2 = 1 or (y2)/a2 – (x2)/b2 =1

Level Surfaces: Paraboloids: F (X; Y) = X2 + Y2 IF Z IS ISOLATED IT TURNS OUT TO BE 0 = X2 + Y2 – Z

Hyperbole of leafs in revolution -1= X2 + Y2 – Z2

ESPHERE: r2 = x2 + y2 +z2

Ellipsoid: 1= (X2) /a2 + (Y2) /b2 + z2/c2

Notice that that inside the sphere or circumference, the formula is still the same tan in the ellipse or ellipsoid, but a=b=c and is the ratio=a=b=c Another thing comes out when (x-h), (y-j), (z-k) is written, it means that the more is rested, the more it gets to the center. Ari Cardenas. (2014/05/27). Curvas y superficies de nivel , Recuperado de aricc.v.blogstop.com/2014/05/curvas-y-supercies-de-nivel.html An example: F(x,y)= 9-x2 – y2 & D= {(x;y); x2 +y2 <= 9} it is also important to remember that the equation of the circumference is: X2 + y2 = r2 Then for the equation D= {(x;y); x2 +y2 <= 9} the ratio equals 3 The z= 9-x2 –y2 is represented as level curves. Z= K as a variable to define it as an arbiter number K = 9-x2 – y2 X2 + y2 = 9 – k} here is where he circumference comes out and is showed as level curves The circumference is taken when this one is equals 0 K=0 X 2 + Y2 = 9

These denotes that the ratio is 3, and it means that in each quadrant from the origin, this will only reach the number 3

Then is well known that when K=0, the circumference has diameter 6 and ratio 3 The next slice: it has to be spotted the amount of wanted curves, that is to say, the K’s arbiter values. Then the next shows off when K=1: X2 + y2 = 9 – 1 X2 + y2 = 8 The last equation solves the diameter, now to find the ratio what must be done is to figure the square root out from the diameter √8 = 2√2 ¿how can this be drawn in a graph?

When k=0 the r=3 When k=1 the r=22√2 = 2,8 It must be spotted that the smaller the gap is between each K, the closer the curves are going to be from each other. Somehow, it is noticeable that in the formula it stays like K less 9, what actually means that:

Every negative value will show off a positive one as a result If K=9 the ratio will be 0 due to the formula 9-9 Except for the number 9, the ratio is infinite when it comes to K “Profesor Particular Puebla (2016/05/19) Curvas de nivel | calculo multivariable [archive de video]. Saved from: Http://Youtube.com/watch?v-5t9trbvx1hw” Multivariable calculus graph and its level curves Like in the one variable functions case, it can be studied the behavior of a two variable function by drawing the graph, the graph of one function of two variables is the set of all its points (X,Y,Z) for those Z= F(x,y) and (x,y) is in the dominium of F. Another way to visualize a two variables function is like a scalar camp that assigns the point (X; Y) the scalar Z= (X; Y) A scalar camp has a characteristic and it is that the F (X; Y) is constant. Example: A meteorological map shows the level curves of equal pressure, called isobaric lines; and when they represent the same temperature they’re called isomeric. Main features: All curves are closed by its self. No curve can be divided If at some point the curves close, it spots a cave If the curves are away from each other, it means that there’s a smooth downhill. The dominium of multivariable functions: EXAMPLE: The functions of this calculus are shown off like: 𝐹(𝑥, 𝑦) =

𝑥𝑦 − 5 2√𝑦 − 𝑥 2

In this case it’s about a rational formula, radical one and fractions one. 1. The values bigger than 0 are wanted to define them: √𝑦 − 𝑥 2 > 0 √𝑦 − 𝑥 2 > √0 what was done here was to radicalize the 0 to eliminate the other side root, then: Y-X2 > 0 And it’s from this point that Y is solved. 𝑌 > 𝑥2

The last formula will have to show its dominium by functions, and not for numeric values, it means that if it is drawn the graph, and this is an easy example, of X2 the line is going go up because it is a parabola. Now that the formula goes like D (X, Y) =Y-X2 ¿How is this formula drawn?

The dominium is = y> all values of parabola X2

The analytical proof: (0;-1) 𝐹(0; −1) = 𝐹(0; −1) = 2

−5 √−1

0(−1) − 5 2√−1 − 02

This result will not be available in the graph because it does not fit in it.

Now values that fit inside the parabola (0,2) 𝐹(0; 2) =

0(2) − 5 2√2 − 02

=

−5 2√2

= 1,76

“ProfesorParticularPuebla (2016/05/16) Dominio En Funciones De Dos O Mas Variables [Archivo de Video]. Saved from: Http://Www.youtube.com/watch?v=wd56qQzAhx0