Partial Differentiation and Multivariable Calculus

Vocabulary practice flashcards covering partial derivatives, total differentials, exactness criteria, reciprocity and cyclic relations, the chain rule, and differentiation of implicit functions based on lecture notes by Dr. Nishu Gupta.

Partial Derivative with Respect to $x$

The rate of change of a multivariable function $f(x, y)$ in the positive $x$ direction, defined as $\frac{\partial f}{\partial x} = \lim_{\Delta x \to 0} \frac{f(x+\Delta x, y) - f(x, y)}{\Delta x}$, provided the limit exists.

Partial Derivative with Respect to $y$

The rate of change of a multivariable function $f(x, y)$ in the positive $y$ direction, defined as $\frac{\partial f}{\partial y} = \lim_{\Delta y \to 0} \frac{f(x, y+\Delta y) - f(x, y)}{\Delta y}$, provided the limit exists.

Subscript Notation for Partial Derivatives

A notation such as $\left(\frac{\partial f}{\partial x}\right)_y$ or $\left(\frac{\partial f}{\partial y}\right)_x$, where the subscript explicitly indicates which variable is held constant during differentiation.

General $n$-Variable Partial Derivative

The derivative of $f(x_1, x_2, \ldots, x_n)$ with respect to the variable $x_i$, defined as $\frac{\partial f}{\partial x_i} = \lim_{\Delta x_i \to 0} \frac{f(x_1, \ldots, x_i+\Delta x_i, \ldots, x_n) - f(x_1, \ldots, x_n)}{\Delta x_i}$.

Monge Shorthand Notations ($p, q, r, s, t$)

The standard single-letter symbols used for the first and second partial derivatives of $f(x, y)$: $p = f_x$, $q = f_y$, $r = f_{xx}$, $s = f_{xy} = f_{yx}$, and $t = f_{yy}$.

Equality of Mixed Partial Derivatives

The principle stating that $\frac{\partial^2 f}{\partial x \partial y} = \frac{\partial^2 f}{\partial y \partial x}$ (or $\frac{\partial^2 f}{\partial x_i \partial x_j} = \frac{\partial^2 f}{\partial x_j \partial x_i}$), which holds provided that the second partial derivatives are continuous at the point in question.

Total Differential

The infinitesimal change of a multivariable function, given by $df = \frac{\partial f}{\partial x}dx + \frac{\partial f}{\partial y}dy$ for two variables, or $df = \sum_{i=1}^n \frac{\partial f}{\partial x_i}dx_i$ for $n$ variables.

Total Derivative

The full rate of change of a multivariable function $f(x_1, x_2, \ldots, x_n)$ with respect to one variable $x_1$ when the other variables depend on $x_1$, given by $\frac{df}{dx_1} = \frac{\partial f}{\partial x_1} + \frac{\partial f}{\partial x_2}\frac{dx_2}{dx_1} + \cdots + \frac{\partial f}{\partial x_n}\frac{dx_n}{dx_1}$.

Exact Differential

A differential expression that integrates directly to form a single function, such as $df = x\,dy + y\,dx$, which directly integrates to $f(x, y) = xy + c$.

Inexact Differential

A differential expression that cannot be integrated directly and is not the straightforward differential of any function, such as $df = x\,dy + 3y\,dx$.

Integrating Factor

A suitable function by which an inexact differential is multiplied through in order to convert it into an exact differential.

Exactness Condition (Two Variables)

The necessary and sufficient condition for a differential $df = M(x, y)dx + N(x, y)dy$ to be exact, which is $\frac{\partial M}{\partial y} = \frac{\partial N}{\partial x}$.

Generalized Exactness Condition ($n$ Variables)

The requirement for $df = \sum_{i=1}^n g_i(x_1, \ldots, x_n)dx_i$ to be exact, requiring $\frac{\partial g_i}{\partial x_j} = \frac{\partial g_j}{\partial x_i}$ for all pairs $i, j$, yielding $\frac{n(n-1)}{2}$ conditions.

Reciprocity Relation

The relation $\left(\frac{\partial x}{\partial y}\right)_z = \frac{1}{\left(\frac{\partial y}{\partial x}\right)_z}$, which holds provided both partial derivatives exist and neither is equal to zero.

Cyclic Relation

The relation connecting three interrelated variables $x$, $y$, and $z$: $\left(\frac{\partial x}{\partial y}\right)_z \left(\frac{\partial y}{\partial z}\right)_x \left(\frac{\partial z}{\partial x}\right)_y = -1$, which holds provided none of the derivatives vanish.

Chain Rule for Partial Differentiation

A formula to compute the total rate of change of $f(x, y)$ with respect to a parameter $u$ when $x = x(u)$ and $y = y(u)$, expressed as $\frac{df}{du} = \frac{\partial f}{\partial x}\frac{dx}{du} + \frac{\partial f}{\partial y}\frac{dy}{du}$.

First Derivative of an Implicit Function

The derivative $\frac{dy}{dx}$ obtained from an implicit functional relation $f(x, y) = 0$, expressed as $\frac{dy}{dx} = -\frac{\partial f / \partial x}{\partial f / \partial y} = -\frac{f_x}{f_y} = -\frac{p}{q}$.

Second Derivative of an Implicit Function

The second derivative $\frac{d^2 y}{dx^2}$ derived from an implicit function $f(x, y) = 0$, given by $\frac{d^2 y}{dx^2} = -\frac{q^2 r - 2pqs + p^2 t}{q^3}$, where $p = f_x$, $q = f_y$, $r = f_{xx}$, $s = f_{xy}$, and $t = f_{yy}$.