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    • Your favorite 3D function

      Zoë Zoë

      I think the following 3D function is pretty cool. It can be written in Cartesian coordinates $$f(x,y)=\sqrt{x^2+y^2}$$ or in Polar coordinates
      $$f(r\cos(\theta),r\sin(\theta))=\sqrt{r^2}.$$

      If you plot it in Mathematica, it looks like an ice cream cone. 

      ParametricPlot3D[{r*Cos[t], r*Sin[t], (r^2)^.5}, {t, 0, 2*Pi}, {r, 0, 
  4}, BoxRatios -> {1, 1, 1}, Ticks -> False]

      mark

    • Exam Question

      lusta lusta

      I was very curious about number 7,

      Prove the every polar azimuthal projection satisfies

      $T_\phi \cdot T_\theta = 0$.

      How can we show for $\textbf{every}$ azimuthal projection

      mark

    • Exam Question

      mega_late mega_late

      Check it 1,2,
      recall that azimuthal projection satisfy the equation $T(\phi,\theta)=(r(\phi)cos(\theta),r(\phi)sin(\theta))$. Then $T_{\phi}=((r'(\phi)cos(\theta),r'(\phi)sin(\theta))$ and $T_{\theta}=((r(\phi)(-sin(\theta)),r(\phi)cos(\theta))$. Then $T_{\phi} \cdot T_{\theta} = ((r'(\phi)cos(\theta),r'(\phi)sin(\theta)) \cdot ((r(\phi)(-sin(\theta)),r(\phi)cos(\theta)) = (r'(\phi)cos(\theta))(r(\phi)(-sin(\theta))) + (r'(\phi)sin(\theta))(r(\phi)cos(\theta)) = -(r'(\phi)cos(\theta))(r(\phi)(sin(\theta))) + (r'(\phi)sin(\theta))(r(\phi)cos(\theta)) = 0$

      mark

    • Another Exam Question

      mega_late mega_late

      Prove that every cylindrical map projection satisfies $T_{\phi} \cdot T_{\theta}=0$

      mark

    • Sahil wants points

      SighHillDoveN SighHillDoveN

      Using the code: 

      heatSystem = {
   D[u[x, t], t] == D[u[x, t], x, x],
   u[x, 0] == Sin[5 x],
   u[0, t] == 0, u[1, t] == 1};
u[x_, t_] = NDSolveValue[heatSystem,
  u[x, t], {x, 0, 1}, {t, 0, 1}]
Manipulate[Plot[u[x, t], {x, 0, 1},
  ColorFunction -> "TemperatureMap", 
  ColorFunctionScaling -> False,
  PlotRange -> {-1, 1}],
 {t, 0, 0.15}]
pics = Table[Plot[u[x, t], {x, 0, 1},
    ColorFunction -> "TemperatureMap", 
    ColorFunctionScaling -> False,
    PlotRange -> {-1, 1}],
   {t, 0, 0.5, 0.01}];
Export["anim.gif", pics]
u[0.5, 1]

      The graph of the solution is:

      The solution to the system $u_{t}=u_{xx}, u(x,0)=sin(5x), u(0,t)=0, u(1,t)=1$ at the midpoint when t=1 is:

      $u(0.5,1)=0.0894617$

      mark

    • Sahil wants points

      TheMeansOfProduction TheMeansOfProduction

      Use Mathematica to find a numerical solution to the following system: $u_t=u_{xx}$, $u(x,0)=\sin(5x)$, $u(0,t)=0$, $u(1,t)=1$.

      mark

    • Laplace's Equation and Steady State

      NegativeOneTwelfth NegativeOneTwelfth

      The equation is as follows.

      $$u=2x-1$$

      Notice that $u(0)=-1$, $u(2)=3$, and $\nabla^2u=0$.
      The graph is quite simple

      If we think of the object as a bar, the $u$ value on the graph show the temperature of the bar at every point on the bar. This is a steady state heat flow equation because the temperature will not change with time.

      mark

    • Laplace's Equation and Steady State

      SighHillDoveN SighHillDoveN

      Find the solution to the following system consisting of Laplace's equation with boundary conditions:

      $\nabla^2u=0, u(0)=-1,u(2)=3$

      Your solution should be quite a simple algebraic expression. Graph the solution and interpret it in the context of steady state heat flow.

      mark

    • Green's Theorem #3

      Red Red

      So basically we are given $$\int_C F\cdot dn=\int \int divF=\int \int 2x+3y$$ It is a lot easier to do this in polar coordinates so we have $x=0.1cos(\theta)$ and $y=0.1sin(\theta)$ thus $divF=0.2cos(\theta)+0.3sin(\theta)$. This just becomes a matter of integration so:

      $\int_C F\cdot dn= \int_{0}^{2\pi} \int_{0}^{1} 0.2cos(\theta)+0.3sin(\theta) \space drd\theta=$

      $\int_{0}^{2\pi} 0.2cos(\theta)+0.3sin(\theta) \space d\theta=0.2 sin(\theta)-0.3cos(\theta)=0$

      mark

    • Green's Theorem #3

      SighHillDoveN SighHillDoveN

      Suppose that $\text{div}\overrightarrow{F} = 2x + 3y$. Find the approximate flux of a vector field across a circle of
      radius 0.1 centered at the point (1, 1).

      mark