<aside> <img src="https://s3-us-west-2.amazonaws.com/secure.notion-static.com/57cc4d1d-c515-4dc0-977e-849a019146e4/lagrange_multipliers.png" alt="https://s3-us-west-2.amazonaws.com/secure.notion-static.com/57cc4d1d-c515-4dc0-977e-849a019146e4/lagrange_multipliers.png" width="40px" /> Lagrange Multipliers are used to find local minima and maxima of a scalar field subject to constraint.

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Example:

given a function $f(x,y)$ with constraints $g(x,y)$ where $\text dg=0$. To find the maxima and minima we do:

$$ \begin{aligned}\frac{\partial f}{\partial x} - \lambda\frac{\partial g}{\partial x} & = 0 \\[0.2cm]\frac{\partial f}{\partial y} - \lambda\frac{\partial g}{\partial y} & = 0 \end{aligned} $$

• when $f=\lambda_i$ are the solutions

Div and Curl

$$ \begin{aligned} {\rm div}(\vec{F}) &= \vec{\nabla} \cdot \vec{F} \\[0.2cm] {\rm curl} (\vec{F}) &= \vec{\nabla} \times \vec{F} \end{aligned} $$

• For a cartesian vector field $\vec{F} = F_x(x,y,z)\,\hat{i}+ F_j(x,y,z)\,\hat{j} + F_z(x,y,z)\,\hat{k}$ then:

$$ \begin{aligned} \vec{\nabla} \cdot \vec{F} &= \left( \frac{\partial}{\partial x}\,\hat{i}+ \frac{\partial}{\partial y}\,\hat{j}+ \frac{\partial}{\partial k}\,\hat{k} \right) \cdot \left( F_x\,\hat{i}+ F_y\,\hat{j}+ F_z\,\hat{k} \right) = \left( \frac{\partial F_x}{\partial x}+ \frac{\partial F_y}{\partial y}+ \frac{\partial F_z}{\partial k} \right) \\[0.3cm] \vec{\nabla} \times \vec{F} &= \begin{vmatrix} \hat{i} & \hat{j} & \hat{k} \\ \frac{\partial }{\partial x} & \frac{\partial }{\partial y} & \frac{\partial}{\partial z} \\ F_x & F_y & F_z \\ \end{vmatrix} = \left(\frac{\partial F_z}{\partial y} - \frac{\partial F_y}{\partial z} \right)\,\hat{i} - \left(\frac{\partial F_z}{\partial x} - \frac{\partial F_x}{\partial z} \right)\,\hat{j}+ \left(\frac{\partial F_y}{\partial x} - \frac{\partial F_x}{\partial y} \right)\,\hat{k} \end{aligned} $$

Uses:

• Quantifying the circulation (curl) and flux (div) of a vector field in a region
• Determining if a force is conservative (in this case $\vec \nabla \times \vec F=0$ )
• Simplifying and unifying the laws of electromagnetism in a constant formalism. Gauss', Ampere's and Faraday's Laws can all be expressed in terms of $\vec{\nabla} \cdot \vec{E}, \vec{\nabla} \times \vec{E}, \vec{\nabla} \times \vec{B}$