Operator notations -Trigonometry formulas

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Post published:October 29, 2021
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Operator notations

Gradient:

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

grad (f) = \nabla f = (\frac{\partial}{\partial x}, \frac{\partial}{\partial y}, \frac{\partial}{\partial z}) 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,…,x_n), also called a scalar field, the gradient is the vector field:

\nabla ψ = (\frac{\partial}{\partial x_{1}}, \dots, \frac{\partial}{\partial x_{n}}) ψ = \frac{\partial ψ}{\partial x_{1}} e_{1} + \dots + \frac{\partial ψ}{\partial x_{n}} e_{n}

where e_i are orthogonal unit vectors in arbitrary directions.

For a vector field A=(A₁,…,A_n) 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} = {(\frac{\partial A_{i}}{\partial x_{j}})}_{i j}

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=F_xi+F_yj+F_zk is the scalar-valued function:

div F = \nabla \cdot F = (\frac{\partial}{\partial x}, \frac{\partial}{\partial y}, \frac{\partial}{\partial z}) \cdot (F_{x}, F_{y}, F_{z}) = \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

div A = \nabla \cdot 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 \cdot (B \otimes \hat{A}) = \hat{A} (\nabla \cdot B) + (B \cdot \nabla) \hat{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 \cdot (b a^{T}) = a (\nabla \cdot b) + (b \cdot \nabla) a

Curl:

In Cartesian coordinates, for F=F_xi+F_yj+F_zk the curl is the vector field:

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

= (\frac{\partial F_{z}}{\partial y} - \frac{\partial F_{y}}{\partial z}) i + (\frac{\partial F_{x}}{\partial z} - \frac{\partial F_{z}}{\partial x}) j + (\frac{\partial F_{y}}{\partial x} - \frac{\partial F_{x}}{\partial y}) 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 \times F = ε^{i j k} 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

Δ f = \nabla^{2} f = (\nabla \cdot \nabla) 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:

Δ A = \nabla^{2} A = (\nabla \cdot \nabla) 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,

Δ f = 0

Special notations:

In Feynman subscript notation,

\nabla_{B} (A \cdot B) = A \times (\nabla \times B) + (A \cdot \nabla) 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 (A \cdot B) = A \times (\nabla \times B) + (A \cdot \nabla) 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.