Nabla Operator

This is a personal note with personal understanding.

Notations

Index Notation is included.

Scalars are not bold (a,b,c⋯a,b,c\cdotsa,b,c⋯), while vectors are bold (a,b,c⋯\boldsymbol{a,b,c\cdots}a,b,c⋯)

Sometimes for convenience,
∂x=∂∂x∂y=∂∂y∂z=∂∂z\partial_x = \frac{\partial}{\partial x} \qquad\partial_y = \frac{\partial}{\partial y} \qquad\partial_z = \frac{\partial}{\partial z}∂x=∂x∂∂y=∂y∂∂z=∂z∂

Row vector inner product a⋅b=abT\newcommand{\V}[1]{\boldsymbol{#1}}\V{a}\cdot\V{b} = \V{a}\V{b}^Ta⋅b=abT

Definitions

Operators and functions can be expressed as a row vector.
∇=(∂∂x,∂∂y,∂∂z)\nabla = \Big(\frac{\partial}{\partial x}, \frac{\partial}{\partial y}, \frac{\partial}{\partial z}\Big)∇=(∂x∂,∂y∂,∂z∂)

Gradient of a scalar function is a vector
gradf=∇f=(∂∂x,∂∂y,∂∂z)f=(∂f∂x,∂f∂y,∂f∂z)\text{grad} f = \nabla f = \Big(\frac{\partial}{\partial x}, \frac{\partial}{\partial y}, \frac{\partial}{\partial z}\Big)f = \Big(\frac{\partial f}{\partial x}, \frac{\partial f}{\partial y}, \frac{\partial f}{\partial z}\Big)gradf=∇f=(∂x∂,∂y∂,∂z∂)f=(∂x∂f,∂y∂f,∂z∂f)

Divergence of a vector function is a scalar
divF=∇⋅F=∂Fi∂xi=∂Fx∂x+∂Fy∂y+∂Fz∂z\newcommand{\V}[1]{\boldsymbol{#1}} \text{div}\V{F} = \nabla\cdot\V{F} = \frac{\partial F_i}{\partial x_i} = \frac{\partial F_x}{\partial x} + \frac{\partial F_y}{\partial y} + \frac{\partial F_z}{\partial z}divF=∇⋅F=∂xi∂Fi=∂x∂Fx+∂y∂Fy+∂z∂Fz

Curl of a vector function is a vector
curlF=∇×F=det⁡∣e^1e^2e^3∂x∂y∂zFxFyFz∣\newcommand{\V}[1]{\boldsymbol{#1}} \text{curl}\V{F} = \nabla\times\V{F} = \det\left|\begin{matrix}\hat{e}_1 & \hat{e}_2 & \hat{e}_3 \\ \partial_x & \partial_y & \partial_z \\ F_x & F_y & F_z \end{matrix}\right|curlF=∇×F=det∣∣∣∣∣∣e^1∂xFxe^2∂yFye^3∂zFz∣∣∣∣∣∣

Jacobian Matrix of a vector function is a 2d matrix
JF=DF=∇F=(∂xFx∂yFx∂zFx∂xFy∂yFy∂zFy∂xFz∂yFz∂zFz)\newcommand{\V}[1]{\boldsymbol{#1}} \V{J}_\V{F} = D\V{F} = \nabla\V{F} = \left(\begin{matrix}\partial_x F_x & \partial_y F_x & \partial_z F_x \\ \partial_x F_y & \partial_y F_y & \partial_z F_y\\ \partial_x F_z & \partial_y F_z & \partial_z F_z \end{matrix}\right)JF=DF=∇F=⎝⎛∂xFx∂xFy∂xFz∂yFx∂yFy∂yFz∂zFx∂zFy∂zFz⎠⎞

Derivative

Infinitesimal changes form column vectors.

For a scalar function fff, total derivative is
df=(∇f)dr=(∂f∂x,∂f∂y,∂f∂z)[dxdydz]\newcommand{\V}[1]{\boldsymbol{#1}} df = (\nabla f)d\V{r} = \Big(\frac{\partial f}{\partial x}, \frac{\partial f}{\partial y}, \frac{\partial f}{\partial z}\Big)\left[\begin{aligned}dx \\ dy\\ dz\end{aligned}\right]df=(∇f)dr=(∂x∂f,∂y∂f,∂z∂f)⎣⎢⎡dxdydz⎦⎥⎤

For a vector function F\boldsymbol{F}F
dF=(∇F)dr=(∂xFx∂yFx∂zFx∂xFy∂yFy∂zFy∂xFz∂yFz∂zFz)[dxdydz]\newcommand{\V}[1]{\boldsymbol{#1}} d\V{F} = (\nabla \V{F})d\V{r} = \left(\begin{matrix}\partial_x F_x & \partial_y F_x & \partial_z F_x \\ \partial_x F_y & \partial_y F_y & \partial_z F_y\\ \partial_x F_z & \partial_y F_z & \partial_z F_z \end{matrix}\right)\left[\begin{aligned}dx \\ dy\\ dz\end{aligned}\right]dF=(∇F)dr=⎝⎛∂xFx∂xFy∂xFz∂yFx∂yFy∂yFz∂zFx∂zFy∂zFz⎠⎞⎣⎢⎡dxdydz⎦⎥⎤

This can be applied to gradient of dot product.
∇(A⋅B)=A⋅(∇B)+B⋅(∇A)\newcommand{\V}[1]{\boldsymbol{#1}} \nabla(\V{A}\cdot\V{B}) = \V{A}\cdot(\nabla\V{B}) + \V{B}\cdot(\nabla\V{A})∇(A⋅B)=A⋅(∇B)+B⋅(∇A)

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Nabla Operator

Notations

Definitions

Derivative

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