Total Differential


If $f:\mathbb{R}^n \to \mathbb{R}$ is given by $z = f(x_1,x_2,...,x_n)$ and its first order partial derivatives $\frac{\partial f}{\partial x_1},\frac{\partial f}{\partial x_2},...,\frac{\partial f}{\partial x_n}$ exist, the total derivative of $z$ is, $\begin{equation}\label{eq:tdd} dz=\frac{\partial f}{\partial x_1} dx_1 + \frac{\partial f}{\partial x_2} dx_2 +...+\frac{\partial f}{\partial x_n}dx_n \tag{1}\end{equation}$



Total differential of a multivariable function is used in cases where one input variable can affect the outcome both - independently and through its effect on another input variable. It is effective for finding the total change in the value of output for a small change in the values of the input variables.

For instance, in thermodynamics, the pressure of the gas $(p)$ is dependent on the number of moles $(n)$, the volume of the gas $(V)$ and the temperature of the gas ($T$), the corresponding equation being $$ p(n, V, T) = \frac{nRT}{V}$$ where the total change in the value of $p$ for small changes in the values of $n$,$v$,$T$ is given by $$dp = (\frac{\partial p}{\partial n})_{V,T}\hspace{4mm} dn + (\frac{\partial p}{\partial V})_{n,T}\hspace{4mm}  dV+ (\frac{\partial p}{\partial T})_{n,V}\hspace{4mm} dT $$ $$ \implies dp= \frac{RT}{V}dn - \frac{nRT}{V^2}dV+ \frac{nR}{V}dT$$

One of the prime applications of total differential is approximation, which will be discussed in the following sections below.


Bird's Eye View

For any two points $P_1(x_1,y_1)$ and $P_2(x_2,y_2)$ lying on the surface of the function $z=f(x,y)$, the change in the value of the function on moving from $P_1$ to $P_2$ can be found out using the definition of total differential (assuming that the distance between the points $P_1$ and $P_2$ is very very small).



Video 1: Visualization of dz

Context of Definition

According to the equation ($\ref{eq:tdd}$) the total differential of $z$ (dependent variable) denoted by $dz$ is the total change in $z$. 

$\frac{\partial f}{\partial x_1}$ is the change in $z$ per unit change in $x_1$ and $dx_1$ is the differential of $x_1$ (change in $x_1$).

Similarly for $n^{th}$ input, $\frac{\partial f}{\partial x_n}$ is the change in $z$ per unit change in $x_n$ and $dx_n$ is the differential of $x_n$ (change in $x_n$).

So, for a two-variable function $z=f(x,y)$, $dz$ is the sum of the partial derivative of $z$  w.r.t  $x$ times $dx_1$ and partial derivative of $z$  w.r.t  $y$ times $dy$. $$dz = \frac{\partial f}{\partial x}dx + \frac{\partial f}{\partial y}dy$$ 



Video 2: Differentials


Video 3: Total differential of z

Approximation using total differential

Consider the function $z=f(x,y)$, if $\Delta x$ and $\Delta y$ are the increments in $x$ and $y$ respectively, then the corresponding increment of $z$ is $$\Delta z = f(x+\Delta x,y+\Delta y)-f(x,y)$$ Thus the increment $\Delta z$ represents the change in the value of $z$ when $(x,y)$ changes to $(x+\Delta x,y+\Delta y)$ 

Suppose the values of $\Delta x$ and $\Delta y$ are small, then $\Delta z \approx dz$. This is useful in estimating the value of $f(x_0+\Delta x,y_0+\Delta y)$ when $f(x_0,y_0)$ is known: $$f(x_0+\Delta x,y_0+\Delta y)\approx f(x_0,y_0)+dz$$

where $dz = \frac{\partial f}{\partial x}(x_0,y_0)\hspace{3mm}dx+ \frac{\partial f}{\partial y}(x_0,y_0)\hspace{3mm}dy$, $\hspace{5mm}\Delta x = dx$ and $\Delta y = dy$ ($\Delta x$ and $\Delta y$ are small changes)



Video 4: Total differential change

To move from a point $(x_1,y_1)$ to a nearby point $(x_2,y_2)$ on the surface of the function $z=f(x,y)$, let's take a tangent plane to the surface at $(x_2,y_2)$. Now, according to the tangent plane equation, $$\begin{equation} \label{eq:tpa} z = f(x_2,y_2)+f_x(x_2,y_2)(x-x_2)+f_y(x_2,y_2)(y-y_2) \tag{2}\end{equation}$$

This equation is linear in $z$,$x$,$y$, and depends on the partial derivatives w.r.t both in $x$ and $y$ direction. 

Since the points $(x_1,y_1)$ and $(x_2,y_2)$ are nearby, $x_1-x_2 = \triangle x$ and $y_1-y_2 = \triangle y$. So equation($\ref{eq:tpa}$) becomes: $$f(x_1,y_1) = f(x_2,y_2)+f_x(x_2,y_2)\triangle x+f_y(x_2,y_2)\triangle y$$ $$\implies f(x_1,y_1) = f(x_2,y_2)+dz$$



Video 5: Total differential approximation

When is an expression a total differential?

To determine if an expression $g(x,y)\hspace{2mm}dx+h(x,y)\hspace{2mm}dy$ is a total differential of some function $f(x,y)$, it must satisfy the following conditions

  • $p(x,y) = \frac{\partial f}{\partial x}$ and $q(x,y) = \frac{\partial f}{\partial y}$
  • $\frac{\partial^2 f}{\partial x \partial y} = \frac{\partial^2 f}{\partial y \partial x}$ ie $p_y=q_x$



  • Integration by Substitution :

            It states that the integral of $f(u)\frac{du}{dx}$ with respect to $x$ is the integral of $f(u)$ with respect to $u$. Thus: $$\int f(u)\frac{du}{dx}dx = \int f(u) du = F(u)+c$$

  • Finding total derivative from total differential:

            For a given function: $z = f(x_1,x_2,...,x_n)$, total differential of $z$, $dz= \frac{\partial z}{\partial x_1}dx_1+\frac{\partial z}{\partial x_2}dx_2+...+\frac{\partial z}{\partial x_n}dx_n$

$$dz = f_1dx_1+f_2dx_2+...+f_ndx_n$$

            Total derivative of $z$ w.r.t $x_1$ is found by dividing both sides by $dx_1$ $$\frac{dz}{dx_1}= f_1+f_2\frac{dx_2}{dx_1}+...+f_n\frac{dx_n}{dx_1}$$

  • Total differential is used in solving thermodynamic problems
  • The concept of the total differential is used in the exterior derivative.



Differentials were introduced by Leibnitz in the 1680s to denote quantities that are infinitesimal with respect to ordinary quantities but which wear definite ratios to one another- just as ordinary numbers and magnitudes do. For more than a century after the invention of differential calculus, differential equations were treated as closely as analogous to ordinary algebraic equations. In 1684 Leibniz published details of his differential calculus in Nova Methodus pro Maximis et Minimis and in a journal Itemque Tangentibus in Acta Eruditorum established in Leipzig two years earlier. The paper contained the familiar $d$ notation, the rules for computing the derivatives of powers, products, and quotients.

In the nineteenth century, A. L. Cauchy added much rigor to the treatment of both derivatives and differentials. Following in the footsteps of Leibnitz and Lagrange, Cauchy showed how total differential could be determined and found independently of derivatives as the principal part of the increment of a function, although he did not use this term, K. Weierstrass introduced it.



Pause and Ponder

  • For a multivariable function $z = f(x_1,x_2,...,x_n)$, the change in value of $z$ is dependent on the changes in the values of $x_1$,$x_2$,...,$x_n$
  • Why first-order partial derivatives are considered for finding the total differential








Further Reading



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