To determine.To fill: a) The blank in the statement "Del operator: ∇=?" b) The blank in the statement "Gradient: ∇f=?" c) The blank in the "Laplacian: ∇2f=?"

a) In Cartesian coordinate syste, Rn with coordinates (x1,…,xn) and standard basis {e→1,…,e→n}, ∇ is defined in terms of partial derivative operators as: ∇=∑i=1nei∂∂xi =(∂∂x1,…,∂∂xn However, del can also be expressed in other coordinate systems. Therefore, the complete statement is "Del operator: ∇=∑i=1nei∂∂xi". b) In Cartesian coordinate system Rn with coordinates ((x1,…,xn) and respective partial derivatives: ∇f=∂∂xf(x) =[∂f(x)∂x1,…,∂f(x)∂xn]T Therefore, the complete statement is "Gradient: ∇f=[∂f(x)∂x1,…,∂f(x)∂xn]T". c) Laplacian operator is defined as: ∇2f=∂2f∂x2+∂2f∂y2+∂2f∂z2 Therefore, the comlete statement is "Laplacian: ∇2f=∂2f∂x2+∂2f∂y2+∂2f∂z2.

Solved: "To determine.To fill: a) The blank in the..."

College
Algebra
Linear algebra
Alternate coordinate systems

Tabansi

Answered question

2020-12-16

To determine.To fill:
a) The blank in the statement "Del operator:

\nabla = ?

"
b) The blank in the statement "Gradient:

\nabla f = ?

"
c) The blank in the "Laplacian:

\nabla^{2} f = ?

Answer & Explanation

jlo2niT

Skilled2020-12-17Added 96 answers

a)
In Cartesian coordinate syste, $R^{n}$ with coordinates $(x_{1}, \dots, x_{n})$ and standard basis
${{\vec{e}}_{1}, \dots, {\vec{e}}_{n}},$
$\nabla$ is defined in terms of partial derivative operators as:
$\nabla = \sum_{i = 1}^{n} e_{i} \frac{\partial}{\partial x_{i}}$
$= (\frac{\partial}{\partial x_{1}}, \dots, \frac{\partial}{\partial x_{n}}$
However, del can also be expressed in other coordinate systems.
Therefore, the complete statement is "Del operator: $\nabla = \sum_{i = 1}^{n} e_{i} \frac{\partial}{\partial x_{i}}$ ".
b)
In Cartesian coordinate system $R^{n}$ with coordinates $((x_{1}, \dots, x_{n})$ and respective partial derivatives:
$\nabla f = \frac{\partial}{\partial x} f (x)$
$= {[\frac{\partial f (x)}{\partial x_{1}}, \dots, \frac{\partial f (x)}{\partial x_{n}}]}^{T}$
Therefore, the complete statement is "Gradient: $\nabla f = {[\frac{\partial f (x)}{\partial x_{1}}, \dots, \frac{\partial f (x)}{\partial x_{n}}]}^{T}$ ".
c)
Laplacian operator is defined as:
$\nabla^{2} f = \frac{\partial^{2} f}{\partial x^{2}} + \frac{\partial^{2} f}{\partial y^{2}} + \frac{\partial^{2} f}{\partial z^{2}}$
Therefore, the comlete statement is "Laplacian: $\nabla^{2} f = \frac{\partial^{2} f}{\partial x^{2}} + \frac{\partial^{2} f}{\partial y^{2}} + \frac{\partial^{2} f}{\partial z^{2}} .$

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