Can someone please explain why the graph is linear from the start to the surface?
22310
Would the effect of gravity on me change if I were to dig a very deep hole and stand in it? If so, how would it change? Am I more likely to be pulled downwards, or pulled towards the edges of the hole? If there would be no change, why not?
50117
This is my question:  Let P be a plane considered as a surface in 3-space. Show that its second fundamental form is zero. Conversely, show that if the second fundamental form of a surface is identically zero then the surface is a plane.  So far I have that the general equation of the plane is $$ax+by+cz+d=0.$$ Then $$M(u,v)=(u,v,\alpha u + \beta v + \gamma),$$ where $a,b,c,d,\alpha,\beta, \gamma$ are all constants and $c\not=0$. Then $M_u=(1,0,\alpha)$ and $M_v=(0,1,\beta)$. I know that $\vec{n} =\frac{M_u \times M_v}{|M_u \times M_v|}$ which gives $$ \vec{n}=\frac{(-\alpha,-\beta, 1)}{\sqrt{\alpha^2+\beta^2+1}}.$$ Hence, the unit norm is a constant. $$ M_{uu}=M_{uv}=M_{vv}=0$$ Hence $l=m=n=0$, so the second fundamental form is $$ ldu^2+mdudv+ndv^2=0.$$  Now I need to show the converse, but I don't know how. Please help :)
46638
how can I draw the graph for this   the graph of $x \geq 5$ and $y \geq5$ is?  please guide me to draw it   thanks in advance
109244
I'm trying to graph the following equation     $$y=\lim\limits_{n \to \infty}{\sqrt[n]{1+x^n}}$$   ($x \geq  0$)   $x = 0$: $y = \lim\limits_{n \to \infty}{\sqrt[n]{1+0^n}} = \lim\limits_{n \to \infty}{\sqrt[n]{1}} = 1$  $0 &lt; x &lt; 1$: $y = \lim\limits_{n \to \infty}{\sqrt[n]{1+\frac{1}{\alpha}^n}} = \lim\limits_{n \to \infty}{\sqrt[n]{1}} = 1$  $1 \leq x$: $y = \lim\limits_{n \to \infty}{\sqrt[n]{\alpha}} = 1$  It seems like $y = 1$ all the time.  However, shouldn't there also be an asymptote $y=x$? As $x \to \infty$: $y = \lim\limits_{n \to \infty}{\sqrt[n]{x^n}} = x$, right?  If I am to come to the conclusion that the graph is equivalent to $y=1$ and that the asymptote is to be ignored, I must prove that $\lim\limits_{n \to \infty}{\sqrt[n]{1+x^n}} = 1$. Is this equality true for $x \geq 0$?
74095
I need to be able to plot pixels in this pattern. To me, it looks like a sine wave pattern that is both diagonal and convergent.      What would a function for that look like?  Thanks.
80059
From this , I now understand the gradient to be the directional derivative that returns the steepest slope at a point. However, reading my textbooks, they all say that the gradient is normal to the tangent plane. In the above SE question, the gradient sounds like it should be parallel to a direction in the $xy$ plane, not actually up on the surface.   My question is therefore do I have a misunderstanding of the tangent plane, gradient or directional derivative?   In addition, I question how the gradient always passes through the origin or the $xy$ plane? If for instance in the graphic, the gradient had been another direction (say horizontal), then it wouldn't pass through the origin! In addition, the gradient of other points in the graphic would satisfy this example. Many thanks!
87167
I've been working on some animations recently, and I've noticed that I'm having poor luck with certain things, namely the fact that I don't really use Blender's graph editor very well.  One of the major issues that I have is pretty widespread among 3D graphics programs; when I create an animated object, it defaults to using a curved graph instead of a linear graph, which means that the most movement is at the middle of the animation, and stuff just practically stops near the beginning or the end. While this helps with organic camera movements or such, it's also not what I usually want. Can I set animation curves to default to linear?
11817
The graph of $xy = 1$ in $\Bbb C^2$ is connected. True or false?  I know that it is not connected in $\Bbb R^2$, but what is the case of $\Bbb C^2$?
113289
I am currently working on a model which takes two parameters and produces a measurement statistic. Think of it as Z = f(X,Y).   Z is a matrix of my statistics and I am creating a surface plot of it in matlab. Basically, I am looking for a mathematical/analytical way of determining if the surface is smooth, or if it is jagged. Do large values tend to be clustered together or are they dispersed throughout the matrix? - that is my question. Basically, how mixed up are the values of my matrix?  I need to run the model over different parameter sets and I want to be able to analytically determine which one of my surfaces is the smoothest, has the greatest clustering of large values, and ideally, has no negative values.   Any help will be greatly appreciated and please let me know if you need any further information.   Cheers
35551
I hear so many terms involving the word "linear". Linear function, linear equation, linear system, linear operator, linear transformation, linear mapping, linear space, linear algebra, linear electrical circuits, linear filters, linear electrical elements, linear approximation, linear optimization.  I'm getting crazy trying to understand the application of the "linear concept" to all this aspects (function, equation, mapping, system, operator, transformation, algebra, etc.) and I wish to know the one essence that is to be linear. What something has to be to be linear? If I say something is linear, what do I know for sure about that something (no matter what the something is)?       I heard a definition of linearity is by homogeinity (scaling the input results in a scaled output) and addition (summing the inputs results in summing the outputs). Can I apply this simple definition to all the branches (operator, mapping, system, transformation, algebra, ...) I mentioned ? Do they all behave like a line ?  y = ax + b, for example, is a line but doesn't behave like a line because y is not linear.
42402
Drawing the graph of a function is not intuitive. For example, the function $f(x)=x^2$ is special, we can say that given two points of the function the straight line that join those points is above the function. So we write  $$ x_2&lt;(x_1+x_0)(x-x_0)+x^2  $$ How did I obtain the part of the right?  $$ (Y-x_0^2)/   (x-x_0)  =     (x_1^2 -  x_0^2)   /      (x_1 -   x_0) $$ Now I just isolate Y in the equation above.  My question is, can I do the same with $f(x)=x^3$?  And how do I prove it? Can I say that $x^3$ is below the straight line that join two points of the function $x^3$? If and only if $x$ is positive.
93869
Hello i have the following graph:    I'm trying to figure out if it's planar or not. I think it is not planar but i can't find a subgraph that is a subdivision of K3,3 or K5 , to use the kuratowski theorem.  p.s. i can only redraw the graph to show that it is planar or prove that       its not planar with kuratowski theorem. NO other methods allowed.
81453
OK, I've plotted these graphs but am unable to stretch the second one, so that they both have the same values on the vertical lines drawn from one to another. I've tried to set domain=0:2200 for both but it doesn't work.
83559
In my graph theory book exercise, I found a problem that:     Prove that the World is not flat using Mathematics     This picture is used in the exercise, but no idea of applying it.  I would have used the angles don't add to $180^\circ$, but the author is using a fourth city.
56482
Can anybody explain as how to plot a graph of $|x| + |y| = 1$. Here $|x|$ and $|y|$ are absolute values.
21714
If you look the at  there is a graph of the Euler phi function up to 1000:   Out of the plot you can see various trend lines that appear to be linear.  Q1) Are these lines actually lines? (ie do all the points that appear to be on one line actually all solutions to some linear eqn?)  Q2) Given that they're lines and not just something that looks a bit like lines can they be described? and what points are on them?   Q3) Are the points that don't appear to be on lines actually just the first points on lines that would only be 'visible' if we zoomed out and evaluated the Euler phi up to (say) 10000 or whatever number is large enough.
99070
my linear algebr textbook defines a linear transformation/map as one that satisfies:     i.  T(u+v)=T(u) +T(v).    ii. T(cu) = cT(u)   However, what is traditionally called a linear function, in non-abstract algebra (or highschool algebra, or whatever it is formally called), namely: f(x) = a + bx is not a linear mapping according to the linear algebra definition, unless a = 0.  Is there an intuitive reason why the first definition is called a linear map, and why you would not call y=1+x a linear map, despite the fact that it defines a straight line on a plane? I could simply take the definitions as given, but it always helps me to have an intuitive understanding of such terminology.