 # To determine.To fill: a) The blank in the statement "Del operator: displaystylenabla=?" b) The blank in the statement "Gradient: displaystylenabla{f}=?" c) The blank in the "Laplacian: displaystylenabla^{2}{f}=?" Tabansi 2020-12-16 Answered
To determine.To fill:
a) The blank in the statement "Del operator: $\mathrm{\nabla }=?$"
b) The blank in the statement "Gradient: $\mathrm{\nabla }f=?$"
c) The blank in the "Laplacian: ${\mathrm{\nabla }}^{2}f=?$"
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a)
In Cartesian coordinate syste, ${\mathbb{R}}^{n}$ with coordinates $\left({x}_{1},\dots ,{x}_{n}\right)$ and standard basis
$\left\{{\stackrel{\to }{e}}_{1},\dots ,{\stackrel{\to }{e}}_{n}\right\},$
$\mathrm{\nabla }$ is defined in terms of partial derivative operators as:
$\mathrm{\nabla }=\sum _{i=1}^{n}{e}_{i}\frac{\partial }{\partial {x}_{i}}$
$=\left(\frac{\partial }{\partial {x}_{1}},\dots ,\frac{\partial }{\partial {x}_{n}}$
However, del can also be expressed in other coordinate systems.
Therefore, the complete statement is "Del operator: $\mathrm{\nabla }=\sum _{i=1}^{n}{e}_{i}\frac{\partial }{\partial {x}_{i}}$".
b)
In Cartesian coordinate system ${\mathbb{R}}^{n}$ with coordinates $\left(\left({x}_{1},\dots ,{x}_{n}\right)$ and respective partial derivatives:
$\mathrm{\nabla }f=\frac{\partial }{\partial x}f\left(x\right)$
$={\left[\frac{\partial f\left(x\right)}{\partial {x}_{1}},\dots ,\frac{\partial f\left(x\right)}{\partial {x}_{n}}\right]}^{T}$
Therefore, the complete statement is "Gradient: $\mathrm{\nabla }f={\left[\frac{\partial f\left(x\right)}{\partial {x}_{1}},\dots ,\frac{\partial f\left(x\right)}{\partial {x}_{n}}\right]}^{T}$".
c)
Laplacian operator is defined as:
${\mathrm{\nabla }}^{2}f=\frac{{\partial }^{2}f}{\partial {x}^{2}}+\frac{{\partial }^{2}f}{\partial {y}^{2}}+\frac{{\partial }^{2}f}{\partial {z}^{2}}$
Therefore, the comlete statement is "Laplacian: ${\mathrm{\nabla }}^{2}f=\frac{{\partial }^{2}f}{\partial {x}^{2}}+\frac{{\partial }^{2}f}{\partial {y}^{2}}+\frac{{\partial }^{2}f}{\partial {z}^{2}}.$