## Operator notations -Trigonometry formulas

### Operator notations

In the three-dimensional Cartesian coordinate variables, the gradient of some function f(x,y,z) is given by:

$\text{grad}\left(f\right)=\nabla f=\left(\frac{\partial }{\partial x},\frac{\partial }{\partial y},\frac{\partial }{\partial z}\right)f=\frac{\partial f}{\partial x}i+\frac{\partial f}{\partial y}j+\frac{\partial f}{\partial z}k$

where i, j, k are the standard unit vectors for the x, y, z-axes.

More generally, for a function of n variables ψ(x1,…,xn), also called a scalar field, the gradient is the vector field:

$\nabla \psi =\left(\frac{\partial }{\partial {x}_{1}},\cdots ,\frac{\partial }{\partial {x}_{n}}\right)\psi =\frac{\partial \psi }{\partial {x}_{1}}{e}_{1}+\cdots +\frac{\partial \psi }{\partial {x}_{n}}{e}_{n}$

where ei are orthogonal unit vectors in arbitrary directions.

For a vector field A=(A1,…,An) written as a 1 × n row vector, also called a tensor field of order 1, the gradient or covariant derivative is the n × n Jacobian matrix:

$\nabla A={J}_{A}={\left(\frac{\partial {A}_{i}}{\partial {x}_{j}}\right)}_{ij}$

The gradient of a tensor field, A of any order k, is generally written as: grad⁡(A)=∇A and is a tensor field of order k + 1.

### Divergence:

In three-dimensional Cartesian coordinates, the divergence of a continuously differentiable vector field F=Fxi+Fyj+Fzk is the scalar-valued function:

$\text{div}F=\nabla ·F=\left(\frac{\partial }{\partial x},\frac{\partial }{\partial y},\frac{\partial }{\partial z}\right)·\left({F}_{x},{F}_{y},{F}_{z}\right)=\frac{\partial {F}_{x}}{\partial x}i+\frac{\partial {F}_{y}}{\partial y}j+\frac{\partial {F}_{z}}{\partial z}k$

The divergence of a tensor field A of non-zero order k is written as

$\text{div}A=\nabla ·A$

and is a contraction to a tensor field of order k − 1. Specifically, the divergence of a vector is a scalar. The divergence of a higher order tensor field may be found by decomposing the tensor field into a sum of outer products and using the identity,

$\nabla ·\left(B\otimes \stackrel{^}{A}\right)=\stackrel{^}{A}\left(\nabla ·B\right)+\left(B·\nabla \right)\stackrel{^}{A}$

where B⋅∇ is the directional derivative in the direction of B multiplied by its magnitude. Specifically, for the outer product of two vectors,

$\nabla ·\left(b{a}^{T}\right)=a\left(\nabla ·b\right)+\left(b·\nabla \right)a$

### Curl:

In Cartesian coordinates, for F=Fxi+Fyj+Fzk the curl is the vector field:

$$\text{curl}F=\nabla \times F=(\frac {\partial}{\partial x},\frac {\partial}{\partial y},\frac {\partial}{\partial z})\times (F_x,F_y,F_z)= \left| \matrix{ i & j & k \cr \frac {\partial}{\partial x} & \frac {\partial}{\partial y} & \frac {\partial}{\partial z} \cr F_x & F_y & F_z } \right|$$
$=\left(\frac{\partial {F}_{z}}{\partial y}–\frac{\partial {F}_{y}}{\partial z}\right)i+\left(\frac{\partial {F}_{x}}{\partial z}–\frac{\partial {F}_{z}}{\partial x}\right)j+\left(\frac{\partial {F}_{y}}{\partial x}–\frac{\partial {F}_{x}}{\partial y}\right)k$

where i, j, and k are the unit vectors for the x-, y-, and z-axes, respectively.

In Einstein notation, the vector field F=(F1 F2 F3)has curl given by:

$\nabla ×F={\epsilon }^{ijk}{e}_{i}\frac{\partial {F}_{k}}{\partial {x}^{j}}$

where ε is the Levi-Civita parity symbol.

### Laplacian:

In Cartesian coordinates, the Laplacian of a function f(x,y,z) is

$\Delta f={\nabla }^{2}f=\left(\nabla ·\nabla \right)f=\frac{{\partial }^{2}f}{\partial {x}^{2}}+\frac{{\partial }^{2}f}{\partial {y}^{2}}+\frac{{\partial }^{2}f}{\partial {z}^{2}}$

For a tensor field, A, the Laplacian is generally written as:

$\Delta A={\nabla }^{2}A=\left(\nabla ·\nabla \right)A$

and is a tensor field of the same order.

When the Laplacian is equal to 0, the function is called a Harmonic Function. That is,

$\Delta f=0$

### Special notations:

In Feynman subscript notation,

${\nabla }_{B}\left(A·B\right)=A×\left(\nabla ×B\right)+\left(A·\nabla \right)B$

where the notation ∇B means the subscripted gradient operates on only the factor B. Less general but similar is the Hestenes overdot notation in geometric algebra. The above identity is then expressed as:

$\nabla \left(A·B\right)=A×\left(\nabla ×B\right)+\left(A·\nabla \right)B$

where overdots define the scope of the vector derivative. The dotted vector, in this case B, is differentiated, while the (undotted) A is held constant.