Given a function \(f(x_1, x_2, ..., x_n)\):

Eq Table

Operation	Equation	Type
gradient \(\nabla f\)	\(\left( \frac{\partial f}{\partial x_1} \vec{v_1} + \frac{\partial f}{\partial x_2} \vec{v_2} + ... + \frac{\partial f}{\partial x_n} \vec{v_n} \right)\)	Vector
curl \(\nabla \times f\)	\(\left( \frac{\partial f_z}{\partial y} - \frac{\partial f_y}{\partial z} \right) \hat{i} - \left( \frac{\partial f_z}{\partial x}- \frac{\partial f_x}{\partial z}\right) \hat{j}+ \left( \frac{\partial f_y}{\partial x} - \frac{\partial f_x}{\partial y}\right) \hat{k}\)	Vector
div \(\nabla \cdot f\)	\(\frac{\partial f_x}{\partial x} + \frac{\partial f_y}{\partial y} + \frac{\partial f_z}{\partial z}\)	Scalar
laplacian \(\nabla^2 f\)	\(\sum_{i=1}^n \frac{\partial^2 f}{\partial x_i^2}\)	Same as input

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Gradient

\[\nabla f = \left( \frac{\partial f}{\partial x_1} \vec{v_1} + \frac{\partial f}{\partial x_2} \vec{v_2} + ... + \frac{\partial f}{\partial x_n} \vec{v_n} \right)\] \[\nabla f = (F_{x_1}, F_{x_2}, ... , F_{x_n})\]

Curl

\[\text{curl}\ F = \left( \frac{\partial f_z}{\partial y} - \frac{\partial f_y}{\partial z} \right) \hat{i} - \left( \frac{\partial f_z}{\partial x} - \frac{\partial f_x}{\partial z} \right) \hat{j} + \left( \frac{\partial f_y}{\partial x} - \frac{\partial f_x}{\partial y} \right) \hat{k}\] \[\text{curl}\ F = \text{det} \begin{vmatrix} \mathbf{i} & \mathbf{j} & \mathbf{k} \\ \frac{\partial}{\partial x} & \frac{\partial}{\partial y} & \frac{\partial}{\partial z} \\ F_x & F_y & F_z \end{vmatrix}\]

Notations:

\[\text{curl}\ F = \nabla \times F = \text{rot}\ F\]

Div

Given a function \(\vec{f}(x, y, z) = f_x \vec{i} + f_y \vec{j} + f_z \vec{k}\):

\[\text{div}\ F = \frac{\partial f_x}{\partial x} + \frac{\partial f_y}{\partial y} + \frac{\partial f_z}{\partial z}\] \[\text{div}\ F = \nabla \cdot F\]

Note: The output is a scalar!

Laplacian

\[\nabla^2 = \sum_{i=1}^n \frac{\partial^2}{\partial x_i^2}\]

Where \(\nabla^2\) is laplacian operator (not to be confused with gradient squared).

\(\nabla^2\) is unique as it returns the same type of the output (scalar, vector in and out).

Say we have a function \(f(x, y, z) = u \vec{i} + v \vec{j} + w \vec{k}\). Then:

\[\nabla^2 f = (\nabla^2 u,\ \nabla^2 v,\ \nabla^2 w)\] \[\nabla^2 f = \left( \frac{\partial^2 u}{\partial x^2} + \frac{\partial^2 u}{\partial y^2} + \frac{\partial^2 u}{\partial z^2} \right) \hat{i} + \left( \frac{\partial^2 v}{\partial x^2} + \frac{\partial^2 v}{\partial y^2} + \frac{\partial^2 v}{\partial z^2} \right) \hat{j} + \left( \frac{\partial^2 w}{\partial x^2} + \frac{\partial^2 w}{\partial y^2} + \frac{\partial^2 w}{\partial z^2} \right) \hat{k}\]

Implicit Derivative

Given \(F(x,y) = 0\). Then:

\[y' = \frac{dy}{dx} = -\frac{F_x}{F_y}\] \[x' = \frac{dx}{dy} = -\frac{F_y}{F_x}\]

This can be expanded to functions with more variables:

Given \(F(x, y, z, t)\):

\[Y_x = -\frac{F_x}{F_y}\] \[Z_t = -\frac{F_t}{F_z}\]

And so on…

For this to work, move the entire function to one side so that the other side is equal to 0.

Directional Derivative

Given a function \(f(x_1, x_2, ..., x_n)\) and a unit vector \(\vec{u} = (v_1, v_2, ..., v_n)\), the directional derivative:

\[D_{\vec{u}} f(x_1, x_2, .., x_n) = v_1 F_{x_1} + v_2 F_{x_2} + ... + v_n F_{x_n}\] \[D_{\vec{u}} f(x_1, x_2, .., x_n) = \vec{u} \cdot \nabla f\]

Note: The output is a scalar!

The maximum value of \(D_{\vec{u}} f\) is when \(\vec{u}\) has the same direction as \(\nabla f\).

Lagrange Multipliers

Given two functions f, g of same dimensions:

\[\nabla f(x_1, x_2, ..., x_n) = \lambda \nabla g(x_1, x_2, ..., x_n)\]

Where \(\lambda\) is an arbitrary constant.

Green

Let C be a simple closed curve with positive, counterclockwise orientation. Let D be the region enclosed by C.

\[\oint_C (P dx + Q dy) = \iint_D \left(\frac{\partial Q}{\partial x} - \frac{\partial P}{\partial y} \right) dA\]

Where \(dA = dy\ dx = dx \ dy\), depending on the integration order.

Stokes

\[\oint_C \vec{F} \cdot d\vec{\Gamma} = \iint_s (\text{curl}\ \vec{F}) \cdot \hat{n}\ dS\]

Where

• \(\vec{F}\) is force field
• \(d\vec{\Gamma}\) is path differential
• \(C\) is a closed curve
• \(\oint_C \vec{F} \cdot d\vec{\Gamma}\) is the work done by a vector field on a closed curve
• \(S\) is any surface bounded by \(C\)

Full story here

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Notation

• Partial derivatives: \(F_x = \frac{\partial F}{\partial x}\)