martes, 16 de septiembre de 2008

tarea 3.....

Calcular la gradiente de la funcion:

\\f(x,y)= 4x^2 - 3xy + y^2\\ f(x)(xy)= 8x -3y\\ f(y)(xy)= -3x +2y\\ \\ \overline{V} = 8x - 3y i -3x + 2y j





2.- \\f(x,y,z)=\frac{x-y}{x-z}


\\f(x)(x,y,z)=\frac{(x+z)Dx(x-y)-(x-y)Dx(x+z)}{(x+z)^2}


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


\\=\frac{x+z -x-y}{x+z^2}

\\= \frac{z+y}{x+z^2} i


\\f(y)(x,y,z) = \frac{(x+z)Dy (x-y)-(y-y)Dy (x+z)}{x+z^2}



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


\\=\frac{-x-z}{x+z^2} j



\\f(z)(x,y,z)=\frac{(x+z)Dz(x-y)(x-y)Dz(x+z)}{x+z^2}


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

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



\overline{V}= \frac{z+y}{(x+z^2}i - \frac{x+z}{(x+z)^2}j + \frac{x+y}{(x+z)^2} k





Calcular la divergencia y el rotacional del campo vectorial F


f(x,y,z) = 6x^2 j + xy^2j


m= 6x^2 \; n= xy^2 \; r=0

divergencia: 12x + 2xy


\frac{{\partial f}}{{\partial x}}= 12x ; \frac{{\partial f}}{{\partial y}}= 2xy ; \frac{{\partial f}}{{\partial z}}= o


Producto cruz.....

rotacional \overline{V} * F \begin{bmatrix}{\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) - (N)\frac{{\partial f}}{{\partial z}}\end{bmatrix} - \begin{bmatrix}\frac{{\partial f}}{{\partial x}}(R) - (M)\frac{{\partial f}}{{\partial z}}\end{bmatrix} + \begin{bmatrix}\frac{{\partial f}}{{\partial x}}(N) - (M)\frac{{\partial f}}{{\partial y}}\end{bmatrix}

\\=(0-0)i -(0-0)j + 12x(xy^2) - 6x^2(2xy)k\\ \\=0i - 0j + (12X^2 - 6x^3y)k


F(x,y,z) = Sen x i + Cos y j + z^2 k

m= Sen x n= Cos y r = z^2

divergencia: Cos x -Sen y + 2z


\frac{{\partial f}}{{\partial x}}= Cos x \frac{{\partial f}}{{\partial y}}= -Sen y \frac{{\partial f}}{{\partial z}}= 2z


producto cruz......


rotacional \overline{V} * F \begin{bmatrix}{\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)- (n)\frac{{\partial f}}{{\partial z}}\end{bmatrix} - \begin{bmatrix}\frac{{\partial f}}{{\partial x}}(r)- (m)\frac{{\partial f}}{{\partial z}}\end{bmatrix} + \begin{bmatrix}\frac{{\partial f}}{{\partial x}}(n)- (m)\frac{{\partial f}}{{\partial y}}\end{bmatrix}

\\=\begin{bmatrix}Sen y(z^2)-2z(Cos y)\end{bmatrix} - \begin{bmatrix}Cosx (z^2)-2z(Senx)\end{bmatrix} + \begin{bmatrix}Cosx(Cosy) - Senx(Seny)\end{bmatrix}
=(Sen y Z^2 - 2Cos y z) i - (Cos xz^2 - 2 Senxz) j + (Cos x Cos y - Sen x Sen y) k



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