Partial Derivatives

Multivariable calculus is the extension of calculus in one variable to calculus in more than one variable.

A study of limits and continuity in multivariable calculus yields counter-intuitive results not demonstrated by single-variable functions.

A partial derivative of a function of several variables is its derivative with respect to a single variable, with the others held constant.

The tangent plane to a surface at a given point is the plane that "just touches" the surface at that point.

For a function $U$ with two variables $x$ and $y$, the chain rule is given as $\frac{d U}{dt} = \frac{\partial U}{\partial x} \cdot \frac{dx}{dt} + \frac{\partial U}{\partial y} \cdot \frac{dy}{dt}$.

The directional derivative represents the instantaneous rate of change of the function, moving through $\mathbf{x}$ with a velocity specified by $\mathbf{v}$.

The second partial derivative test is a method used to determine whether a critical point is a local minimum, maximum, or saddle point.

The method of Lagrange multipliers is a strategy for finding the local maxima and minima of a function subject to equality constraints.

To solve an optimization problem, formulate the function $f(x,y, \cdots )$ to be optimized and find all critical points first.

Finding extrema can be a challenge with regard to multivariable functions, requiring careful calculation.