Fisica II: 09/13/08

1) Calcular el gradiente dela funcion.

a) f (x, y) $=4x^2-3xy+x^2$

$f^{\prime}x(x, y) = 8x-3y+0 = \begin{bmatrix}{8x-3y}\end{bmatrix}i$

$f^{\prime}y(x, y) = 0-3x+2y = \begin{bmatrix}{2y-3x}\end{bmatrix}j$

$\overline{V} = (8x-3y)i + (-3y+2y)j$

b) f (x, y, z) $= \displaystyle\frac{x-y}{x+y}$

Aplicando formula:

$\displaystyle\frac{u}{v} = \displaystyle\frac{V du-Udv}{v^2}$

$f^{\prime}x (x, y, z) =\displaystyle\frac{(x+z)\:\frac{df}{dx}(x-y)- (x-y)\:\frac{df}{dx}(x+z)}{(x+y)^2}$

$=\displaystyle\frac{(x+z)(1)- (x-y)(1)}{(x+y)^2}$

$=\displaystyle\frac{x+z-x+y}{(x+y)^2}$

= $\begin{bmatrix}{\displaystyle\frac{x+y}{(x+z)^2}}\end{bmatrix} i$

$f^{\prime}y (x, y, z) =\displaystyle\frac{(x+z)\:\frac{df}{dy}(x-y)- (x-y)\:\frac{df}{dy}(x+z)}{(x+y)^2}$

$=\displaystyle\frac{(x+y)(-1)-(x-y)(0)}{(x+y)^2}$

$=\begin{bmatrix}\displaystyle\frac{-x-z}{(x+z)^2}\end{bmatrix}j$

$f^{\prime}z (x, y, z) =\displaystyle\frac{(x+z)\:\frac{df}{dz}(x-y)- (x-y)\:\frac{df}{dz}(x+z)}{(x+y)^2}$

$=\displaystyle\frac{(x+z)(0)-(x-y)(1)}{(x+z)^2}$

$=\begin{bmatrix}\displaystyle\frac{-x+y}{(x+z)^2}\end{bmatrix}k$

2) Calcular la Divergencia y el Rotacional de cada vectorial F.

a) F (x, y, z) $= 6x^2i-xy^2j$

$M=6x^2\,i$ $N=xy^2\,j$ $R=0\,k$

Divergencia $=12x\,i-2xy\,j$

$\frac{{\partial f}}{{\partial x}} =12x\,i$ $\frac{{\partial f}}{{\partial y}}= -2xy\,j$ $\frac{{\partial f}}{{\partial z}}= 0\,k$

Rotacional $=\overline{V} * F =\begin{bmatrix}{i}&{j}&{k}\\{\frac{{\partial f}}{{\partial x}}}&{\frac{{\partial f}}{{\partial y}}}&{\frac{{\partial f}}{{\partial z}}}\\{M}&{N}&{R}\end{bmatrix}$

$=\begin{bmatrix}(\frac{{\partial f}}{{\partial y}})(R)-(\frac{{\partial f}}{{\partial z}})(N)\end{bmatrix}i\,-\,\begin{bmatrix}(\frac{{\partial f}}{{\partial x}})(R)-(\frac{{\partial f}}{{\partial z}})(M)\end{bmatrix}j\,+\,\begin{bmatrix}(\frac{{\partial f}}{{\partial x}})(N)-(\frac{{\partial f}}{{\partial y}})(M)\end{bmatrix}k$

$=\,i(0-0)\,-\,j(0-0)\,+\,k(-12x^2y^2+12x^3y)$

$=\,(0)i,\,-\,(0)j,\,(-12x^2y\,(y-x))k$

b) F (x, y, z) $= Senx\,i\,+\,Cosy\,j\,+\,z^2\,k$

$M=Senx\,i$ $N=Cosy\,j$ $R=z^2\,k$

Divergencia = Cosx i - Seny j + 2z k

$\frac{df}{dx}= Cos x\,i$ $\frac{{\partial f}}{{\partial y}} = -Seny\,j$ $\frac{{\partial f}}{{\partial z}}= 2z\,k$

Rotacional $=\overline{V} * F =\begin{bmatrix}{i}&{j}&{k}\\{\frac{{\partial f}}{{\partial x}}}&{\frac{{\partial f}}{{\partial y}}}&{\frac{{\partial f}}{{\partial z}}}\\{M}&{N}&{R}\end{bmatrix}$

$=\begin{bmatrix}(\frac{{\partial f}}{{\partial y}})(R)-(\frac{{\partial f}}{{\partial z}})(N)\end{bmatrix}i\,-\,\begin{bmatrix}(\frac{{\partial f}}{{\partial x}})(R)-(\frac{{\partial f}}{{\partial z}})(M)\end{bmatrix}j\,+\,\begin{bmatrix}(\frac{{\partial f}}{{\partial x}})(N)-(\frac{{\partial f}}{{\partial y}})(M)\end{bmatrix}k$

$=\begin{bmatrix}(-Seny)(z^2)-(2z)(Cosy)\end{bmatrix}i\,-\,\begin{bmatrix}(Cosx)(z^2)-(2z)(Senx)\end{bmatrix}j\,+\,\begin{bmatrix}(Cosx)(Cosy)-(-Seny)(Senx)\end{bmatrix}k$

$=\begin{bmatrix}(-z^2Seny - 2zCosy\end{bmatrix}i\,-\,\begin{bmatrix}z^2Cosx - 2zSenx\end{bmatrix}j\,+\,\begin{bmatrix}CosxCosy + SenySenx\end{bmatrix}k$

c) F (x, y, z) $=e^xSeny\,i\,-\,e^xCosx\,j$

En el Punto (0, 0, 3)

$M= e^xSeny\,i$ $N= e^xCosx\,j$ $R= 0\,k$

Divergencia = $e^xSeny\,-\,e^xCosx$

$\frac{{\partial f}}{{\partial x}}= e^xSeny$ $\frac{{\partial f}}{{\partial y}}= e^xCosx$ $\frac{{\partial f}}{{\partial z}} = 0$

Rotacional $=\overline{V} * F =\begin{bmatrix}{i}&{j}&{k}\\{\frac{{\partial f}}{{\partial x}}}&{\frac{{\partial f}}{{\partial y}}}&{\frac{{\partial f}}{{\partial z}}}\\{M}&{N}&{R}\end{bmatrix}$

$=\begin{bmatrix}(\frac{{\partial f}}{{\partial y}})(R)-(\frac{{\partial f}}{{\partial z}})(N)\end{bmatrix}i\,-\,\begin{bmatrix}(\frac{{\partial f}}{{\partial x}})(R)-(\frac{{\partial f}}{{\partial z}})(M)\end{bmatrix}j\,+\,\begin{bmatrix}(\frac{{\partial f}}{{\partial x}})(N)-(\frac{{\partial f}}{{\partial y}})(M)\end{bmatrix}k$

$=\begin{bmatrix}0\end{bmatrix}i\,+\,\begin{bmatrix}0\end{bmatrix}j\,-\,\begin{bmatrix}-2(e^xSeny\,e^xCosx\,+\,e^xSeny\,e^xCosx)\end{bmatrix}k$

Fisica II

sábado, 13 de septiembre de 2008

Tarea 3 ...Gradiente, Divergencia y Rotacional

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