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<p><img src="http://dmpeli.math.mcmaster.ca/Matlab/Math1J03/LectureNotes/Lecture2_5_files/image014.gif" width="457" height="343"/></p>
<p><a href="http://dmpeli.math.mcmaster.ca/Matlab/Math1J03/LectureNotes/Lecture2_5.htm">Source</a></p>
<h2>Introduction</h2>
<p> Hello it's a me again drifter1! Today we continue with my talk about more advanced stuff of <strong>Mathematical Analysis </strong>by covering different<strong> kinds of functions</strong>: <strong>multi-variablle and vector </strong>functions.</p>
<p> In the previous posts we covered <a href="https://steemit.com/mathematics/@drifter1/mathematics-mathematical-analysis-vectors-lines-and-planes">Vectors, Lines, Planes </a>and also a lot of <a href="https://steemit.com/mathematics/@drifter1/mathematics-mathematical-analysis-advanced-plane-types">Plane Types,</a> and those Planes are of course multi-variable functions...</p>
<p> Anyway, this post will disuss some more things about those multi-variable functions and just to have a little more to talk about we will also give a small introduction of Vector functions, which are another representation of curves (or even planes) in space!</p>
<p>So, without further do, let's get started!</p>
<p><br></p>
<h2>Functions</h2>
<p> Let's first start with functions in general (mostly single-variable ones) so that we have a smooth start and so that you remember what you already know about them...</p>
<p> The basis of the whole Mathematical Analysis branch are Functions for which we even talked about in our first post, our introduction to this series that you can find <a href="https://steemit.com/mathematics/@drifter1/mathematics-mathematical-analysis-introduction-functions">here</a>.</p>
<p>There I cover a lot that can be very useful, but here a small recap...</p>
<blockquote> A function f from a non-zero set A to a non-zero set B is a <strong>rule</strong> f, that <strong>corresponds each element x of set A to only one specific element y of set B</strong>. We symbolize it as <strong>f: A -> B</strong>. We say that "<em>f is a function from A to B</em>". </blockquote>
<p>Some more things:</p>
<ul>
<li>x is the independent variable and y is the dependent variable or image of x (y = f(x))</li>
<li>A is the domain of definition and B is the domain of range</li>
</ul>
<p><strong>ATTENTION</strong>:</p>
<blockquote> A function <strong>can have multiple x's that go to the same y </strong>(think about f(x) = c, where c is a constant), but a function <strong>can not have multiple y's that go to the same x</strong> (or x's can not go to two or more different y's).</blockquote>
<p><br></p>
<p>Some <strong>functions </strong>are:</p>
<ul>
<li>f(x) = x^3 + 5</li>
<li>g(x) = 98</li>
<li>h(x) = 5/x</li>
</ul>
<p><br></p>
<p>You can read about the rest in the introduction, cause that is enough for today's topic...</p>
<p><br></p>
<h2>Dual-variable Functions</h2>
<p>In reality most things are dependent of more then one things...</p>
<p>As we know from Physics, Force is:</p>
<p><em>F = ma</em></p>
<p>and so dependent of two variables m and a!</p>
<p><br></p>
<p>Another great <strong>example </strong>is:</p>
<p><em>V = πr^2h</em></p>
<p> that gives us the volume of a spherical cylinder dependent of it's radius (r) and height (h) multiplied by the constant π.</p>
<p>Also check out <a href="http://www.troup.org/userfiles/929/My%20Files/MS%20Math/8%20Math/8th%20Unit%203/Concept%202/posters_surface_area_and_volume_OT.pdf?id=11070">this</a>.</p>
<p><br></p>
<p>So, <strong>what changes</strong> when we add one more variable into a function?</p>
<ul>
<li>The domain of definition now is a set of ordered pairs s (x, y)</li>
<li>The domain of range f(x, y) is a set of real numbers like it was before</li>
</ul>
<p><br></p>
<p>The <strong>definition </strong>of a dual-variable function looks like this:</p>
<p>Suppose D being the set of ordered pairs of real numbers (x, y).</p>
<p>A real function f of two variables is a rule that corresponds one specific real number:</p>
<p><em>w = f(x , y)</em></p>
<p>for each ordered pair (x, y) of D.</p>
<p>D is called the domain of definition of f and the set of values that w gets is called the domain of range.</p>
<p>The independent variables x and y are also called input variables.</p>
<p>The dependent variable w is called a output variable.</p>
<p><br></p>
<p>So, because of that definition the example's equation can also be written like this:</p>
<p><em>V = f(r, h)</em></p>
<p>so that the independent and dependent variables are better distinguished.</p>
<p><br></p>
<h3>Domain of definition</h3>
<p>So, how do we find this set?</p>
<p> We use the exact same method as in single-variable functions by excluding things that cannot be defined like negative roots (complex numbers) and divisions with zero.</p>
<p>For f(x, y) = root(y - x^2) we of course have:</p>
<p><em>D = [0, ∞)</em></p>
<p>which can be found from:</p>
<p><em>y - x^2 >=0 => y >= x^2 and so y >= 0</em></p>
<p><br></p>
<h3>Bounds</h3>
<p> In the same way as single-variable functons where bounded we again define something like that for more than one variables.</p>
<p>A "set" of a Plane is <strong>bounded </strong>if it can be surrounded by a circle of finite radius.</p>
<p>Else we say that it is not bounded...</p>
<p>Some more things:</p>
<ul>
<li>A point (x0, y0) of a set R of plane xy is an <strong>internal point</strong> of R if it is the center of a circle that is totally a part of R.</li>
<li>A point where each circle drawn of it as a center contains the same amount of points inside and outside of R is called a <strong>boundary point</strong>.</li>
</ul>
<p>The set of all internal points define the <strong>internal space</strong> of a space.</p>
<p>A set is <strong>open </strong>when it contains only the internal points and <strong>closed </strong>if it contains only boundary points.</p>
<p><br></p>
<h2>Functions of 3 or more variables</h2>
<p>Some functions may also be dependent of 3 or even more variables!</p>
<p>For<strong> 3 variables </strong>we can continue with what we said previously by saying that:</p>
<blockquote> A function f of 3 or even more variables is a rule that corresponds each ordered trio (x, y, z) of domain of definition D to one only real number w = f(x, y, z).</blockquote>
<p>The domain of range of course contains all the real values of the output variable w.</p>
<p><br></p>
<p>So, <strong>generalizing for n-variables</strong> we get:</p>
<p> A function of of n-variables is a rule that corresponds to each n-pair of real numbers (x1, x2, ..., xn) one real number w = f(x1, x2, ..., xn).</p>
<p>The variables x1, x2, ..., xn are the independent input variables.</p>
<p>The variable w is the dependent output variable.</p>
<p><br></p>
<p>We can again define things like before, but I will skip it for simplicity...</p>
<p><br></p>
<h2>Plotting those functions</h2>
<p>A function of 1 variable can be plotted very easily in 2-dimensional space.</p>
<p>A function of 2 variables is plotted in 3-dimensional space and so can be represented visually.</p>
<p>But what about the others?</p>
<p><img src="https://upload.wikimedia.org/wikipedia/commons/thumb/5/55/8-cell-simple.gif/255px-8-cell-simple.gif" width="255" height="255"/></p>
<p><a href="https://simple.wikipedia.org/wiki/4D">Source</a></p>
<p> <strong> We can't represent and understand those functions of 3 or more variables </strong>in a way that is more close to us, but there are many strong theorems and tools that help us understand them and use them in mathematical, physics, chemistry etc. problems.</p>
<p><br></p>
<h2>Limit and Continuity of multi-variable Functions</h2>
<p>The limit doesn't change that much...</p>
<p>We define the<strong> limit of a 2-variable function</strong> like that:</p>
<p><em>lim (x,y) -> (x0, y0) [f(x, y)] = L</em></p>
<p>And so <strong>each variable tents to a value</strong>!</p>
<p><br></p>
<p>We again can define one-sided limits and so <strong>the limit only exists when</strong>:</p>
<ol>
<li>The limit is finite (not infinity, but a real number)</li>
<li>The one-sided limits have the same exact value</li>
</ol>
<p><br></p>
<p> For the continuity not much changes again and so <strong>a function f(x, y) is continuous at the point (x0, y0) </strong>when:</p>
<ol>
<li>f can be defined at (x0, y0)</li>
<li>the limit at that point exists AND</li>
<li>lim (x, y) -> (x0, y0) [f(x, y)] = f(x0, y0)</li>
</ol>
<p>A function is <strong>continuous </strong>when this is true for all the points!</p>
<p><br></p>
<p>The same exact stuff can also be said for 3 or more variables...</p>
<p><br></p>
<p>Derivatives, Integrals etc. will be discussed separately in other posts later on...</p>
<p><br></p>
<h2>Vector functions</h2>
<p> So, after talking about multi-variable functions we can now also get into another form of function, which are the vector functions...</p>
<p><br></p>
<p><strong> Curves </strong>in space are defined by functions for each <strong>coordinate </strong>in this space in a specific "period" I of time so that:</p>
<p><em>x = f(t), y = g(t) and z = h(t) with t in I</em></p>
<p>The functions/equations x, y, z are parameters of this curve.</p>
<p> So, those points (x, y, z) = (f(t), g(t), h(t)) compose the curve of an "object" if we think of this curve as the motion of an object...</p>
<p><br></p>
<p>Another way of <strong>representing </strong>that is by using a <strong>vector</strong>:</p>
<p><em>r(t) = OP = f(t)i + g(t)j + h(t)k</em></p>
<p>where O is the center of the axes and P is the final destination P(f(t), g(t), h(t)).</p>
<p>The function f, g, h are called <strong>component functions</strong> of the position vector r.</p>
<p><br></p>
<p>So, what is r(t)?<br>
r(t) is the<strong> vector function</strong> of the real variable t in the interval I.</p>
<p>Generalizing it even more we can say that:</p>
<blockquote> A vector function (function with vector values) with a domain of definition D is a rule that <strong>corresponds a vector in space</strong> for each element of D.</blockquote>
<p> For the time being the vector functions define only curves, but later on (other posts) we will see that we can also define planes!</p>
<p><br></p>
<p>An <strong>example</strong>:</p>
<p><em>r(t) = cos(t)i + sin(t)j + tk</em></p>
<p>This function can desribe a Helix shape like this one:</p>
<p><img src="https://upload.wikimedia.org/wikipedia/commons/thumb/2/29/Helix.svg/220px-Helix.svg.png" width="220" height="322"/></p>
<p><a href="https://en.wikipedia.org/wiki/Helix">Source</a></p>
<p><br></p>
<p>What about <strong>limits and continuity</strong>?</p>
<p>Well, the <strong>limit </strong>will equal to the limit of each "part" and so:</p>
<p><em>lim t->t0 [r(t)] = lim t->t0 [f(t)] i + lim t->t0 [g(t)] j + lim t->t0 [h(t)] k</em></p>
<p>Such a function will be <strong>continuous at the point t = t0 </strong>when:</p>
<p><em>lim t->t0 [r(t)] = r(t0)</em></p>
<p>When that happens for all the points of D then the function is <strong>continuous</strong>!</p>
<p><br></p>
<p>We will talk about derivatives and more stuff another time..</p>
<p><br></p>
<p>And so this is it for today and I hope that you enjoyed it!</p>
<p>Next time we will get into differentiation/derivatives of this types of functions!</p>
<p>Bye!</p>
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