Total differential

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Total differential

Recall that implicit differentiation is commonly used to find derivatives of functions defined implicitly or to find derivatives of functions that cannot be explicitly solved for one variable, y = f(x), dy = f’(x)dx, e.g., y = sin^-1(x) ⇒ x = sin(y) ⇒ dx = cos(y)dy ⇒ $\frac{dy}{dx}=\frac{1}{cos(y)}=\frac{1}{\sqrt{1-x^2}}$

In multivariable Calculus, The total differential of a multivariable function f(x₁, x₂, ···,x_n) is an expression that describes or encodes how a small change in each of its variables affects the function’s value. It is denoted by df and is defined as follows:df = $\frac{\partial f}{\partial x}d_x+\frac{\partial f}{\partial y}d_y+\frac{\partial f}{\partial z}d_z = f_xdx + f_ydy +f_zdz.$

It describes how the function changes as you move along an infinitesimal displacement in each of its variables, hence providing information about the rate of change of the function in each direction Besides, provides an approximation for small variations Δx, Δy, Δz, so Δf ≈ f_xΔx + f_yΔy + f_zΔz.

The Chain Rule for multivariable functions is a fundamental concept in calculus that allows us to compute the derivative of a composite function. It states that if we have a function f(x, y, z) where x = x(t), y = y(t), and z = z(t) are functions of another variable t, then the derivate of f with respect to t is given by $\frac{df}{dt} = f_x\frac{dx}{dt} +f_y\frac{dy}{dt}+f_z\frac{dz}{dt}$.

Proof. (2 versions)

1. Version. df = $f_xdx + f_ydy +f_zdz,$ where x = x(t), y = y(t), and z = z(t) are functions of another variable t⇒ dx = x’(t)dt, dy = y’(t), dz = z’(t)dt ⇒ df = $f_xdx + f_ydy +f_zdz = f_xx’(t)dt + f_yy’(t)dt +f_zz’(t)dt$ ⇒[By dividing by dt] $\frac{df}{dt} = f_x\frac{dx}{dt} +f_y\frac{dy}{dt}+f_z\frac{dz}{dt},$ (The Chain Rule).

2. Version. Δf ≈ $f_xΔx + f_yΔy +f_zΔz$⇒[By dividing by Δt] $\frac{Δf}{Δt} = \frac{f_xΔx + f_yΔy +f_zΔz}{Δt} = f_x\frac{Δx}{Δt} + f_y\frac{Δy}{Δt} + f_z\frac{Δz}{Δt}⇒$ as Δt → 0, the approximation becomes an equality and we get the Chain Rule again, $\frac{df}{dt} = f_x\frac{dx}{dt} +f_y\frac{dy}{dt}+f_z\frac{dz}{dt}.$

Exercises

• w = x²y+z, x = t, y = e^t, and z = sin(t).

Let’s apply the Chain Rule, $\frac{dw}{dt} = 2xy\frac{dx}{dt} + x^2\frac{dy}{dt} + \frac{dz}{dt} = 2te^t + t^2e^t+cos(t)$.

Another way of seeing it is as follows, w(t) = x²y+z = t²e^t + sin(t) ⇒ $\frac{dw}{dt} = 2te^t+t^2e^t+cos(t)$, that is obviously the same answer.

• In particular, let f = uv, where u = u(t) and v = v(t) ⇒ $\frac{d(uv)}{dt}=f_u\frac{du}{dt}+f_v\frac{dv}{dt} = v\frac{du}{dt}+u\frac{dv}{dt}$, and we get the tradition Chain Rule.

• Similarly, let g = ^u⁄_v where u = u(t) and v = v(t) ⇒$\frac{d(u/v)}{dt} =f_u\frac{du}{dt}+f_v\frac{dv}{dt} = =\frac{1}{v}\frac{du}{dt}+\frac{-u}{v^2}\frac{dv}{dt} = \frac{u’v-v’u}{v^2}$, that is, the quotient rule.

• w = f(x, y) where x = x(u, v) and y = y(u, v)

We already know that $dw = f_xdx +f_ydy = f_x(x_udu +x_vdv) +f_y(y_udu +y_vdv) =$[Collecting terms] $(f_xx_u+f_yy_u)du + (f_xx_v + f_yy_v)dv$ ⇒ $\frac{\partial f}{\partial u} = f_xx_u+f_yy_u, \frac{\partial f}{\partial v} = f_xx_v + f_yy_v$, that is, $\frac{\partial f}{\partial u} = \frac{\partial f}{\partial x}\frac{\partial x}{\partial u}+\frac{\partial f}{\partial y}\frac{\partial y}{\partial u}, \frac{\partial f}{\partial v} = \frac{\partial f}{\partial x}\frac{\partial x}{\partial v}+\frac{\partial f}{\partial y}\frac{\partial y}{\partial v}$

• Example. Polar coordinates, x = rcos(θ), y = rsin(θ),

$\frac{\partial f}{\partial r} = \frac{\partial f}{\partial x}\frac{\partial x}{\partial r}+\frac{\partial f}{\partial y}\frac{\partial y}{\partial r} = f_xcos(θ) + f_ysin(θ)$

Bibliography

1. NPTEL-NOC IITM, Introduction to Galois Theory.
2. Algebra, Second Edition, by Michael Artin.
3. LibreTexts, Calculus and Calculus 3e (Apex). Abstract and Geometric Algebra, Abstract Algebra: Theory and Applications (Judson).
4. Field and Galois Theory, by Patrick Morandi. Springer.
5. Michael Penn, and MathMajor.
6. Contemporary Abstract Algebra, Joseph, A. Gallian.
7. YouTube’s Andrew Misseldine: Calculus. College Algebra and Abstract Algebra.
8. MIT OpenCourseWare 18.01 Single Variable Calculus, Fall 2007 and 18.02 Multivariable Calculus, Fall 2007.
9. Calculus Early Transcendentals: Differential & Multi-Variable Calculus for Social Sciences.