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Appendix
 
Next: Questions Up: VECTORS, TENSORS AND MATRIX Previous: Review of Vector and   Chapters

Scientific Workplace Applications

Scientific Workplace can perform many time saving operations.

DOT PRODUCT

$\mathbf{u}=(1,x,y)$

$\mathbf{v}=(1,0,0)$

$\mathbf{u}\cdot\mathbf{v}=1$

CROSS PRODUCT

$\mathbf{u}=(1,2,3)$

$\mathbf{v}=(2,3,4)$

$\mathbf{u}\times\mathbf{v}=\left(-1,2,-1\right) $

SOLVING SYSTEMS OF EQUATIONS

$\begin{array}{l} 2x-3y+4z-8u+2v=-14\\ -x+8y-3z-u+6v=32\\ 6x+7y-2z+u-v=13\\ 20x+3y-6z+6u-4v=12\\ -x-6y-4z-2u-v=-38 \end{array}$, Solution is : $\left\{x=1,y=2,z=3,u=4,v=5\right\}$

This worksheet demonstrates how to compute gradient, divergence, and curl in rectangular coordinate systems. For additional help, see Scientific Workplace's help files.

GRADIENT

Gradient is defined as the del operator operating on a scalar field. Simply stated:


\begin{displaymath}\mathbf{del}\,f=<\frac{\partial}{\partial x}f,\frac{\partial}... ...frac{\partial}{\partial z}f> \text{where f is a scalar field.} \end{displaymath}

Gradient is a vector .

$f(x,y,z)=3x^2+2yz$

$\nabla f(x,y,z)=\begin{array}{c} 6x\\ 2z\\ 2y \end{array}$

DIVERGENCE

Divergence is the dot product of $\mathbf{del}$ and $\mathbf{F}$ where $\mathbf{F}$ is a vector field of the form:
$\mathbf{F}(\,x,y,z\,)=<A(\,x,y,z\,),B(\,x,y,z\,),C(\,x,y,z\,)>$. Simply stated:

\begin{displaymath}\mathbf{Divergence}\,\mathbf{F}=<\frac{\partial}{\partial x},... ...partial}{\partial y}, \frac{\partial}{\partial z}>\cdot<A,B,C> \end{displaymath}

Divergence is a scalar field .

$\mathbf{F}(x,y,z)=(x,y^2,z)$

$\nabla\cdot\mathbf{F}(x,y,z)=2+2y$

CURL

Curl is the cross product of $\mathbf{del}$ and $\mathbf{F}$ where $\mathbf{F}$ is a vector field of the form:
$\mathbf{F}(\,x,y,z\,)=<A(\,x,y,z\,),B(\,x,y,z\,),C(\,x,y,z\,)>$. Simply stated:


\begin{displaymath}\mathbf{Curl}\,\mathbf{F}=<\frac{\partial}{\partial x},\frac{\partial}{\partial y},\frac{\partial}{\partial z}> \times <A,B,C> \end{displaymath}

Curl is a vector field.

$\mathbf{F}(x,y,z)=(x^2,xz,y^2z)$

$\nabla\times\mathbf{F}(x,y,z)=\left( 2yz-x,0,z\right)$


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