Mathematics for Physics I

Definition and Interpretation

Derivatives represent rates of change and are fundamental to understanding how quantities vary with respect to one another in physics.

Formal Definition

The derivative of a function f(x) at a point x = a is defined as:

\[ f'(a) = \lim_{h \to 0} \frac{f(a + h) - f(a)}{h} \]

Alternatively, using the limit notation:

\[ f'(x) = \lim_{\Delta x \to 0} \frac{f(x + \Delta x) - f(x)}{\Delta x} \]

Geometric Interpretation

The derivative f'(a) represents the slope of the tangent line to the curve y = f(x) at the point (a, f(a)).

The equation of this tangent line is:

\[ y - f(a) = f'(a)(x - a) \]

This geometric interpretation provides a visual understanding of the rate of change at a specific point.

Interactive visualization of the derivative as the slope of a tangent line

Basic Differentiation Rules

Power Rule

\[ \frac{d}{dx}[x^n] = nx^{n-1} \]

This rule applies for any real number n.

Constant Multiple Rule

\[ \frac{d}{dx}[cf(x)] = c\frac{d}{dx}[f(x)] \]

Where c is a constant.

Sum and Difference Rules

\[ \frac{d}{dx}[f(x) \pm g(x)] = \frac{d}{dx}[f(x)] \pm \frac{d}{dx}[g(x)] \]

The derivative of a sum is the sum of the derivatives.

Product Rule

\[ \frac{d}{dx}[f(x)g(x)] = f(x)\frac{d}{dx}[g(x)] + g(x)\frac{d}{dx}[f(x)] \]

The derivative of a product is not simply the product of the derivatives.

Quotient Rule

\[ \frac{d}{dx}\left[\frac{f(x)}{g(x)}\right] = \frac{g(x)\frac{d}{dx}[f(x)] - f(x)\frac{d}{dx}[g(x)]}{[g(x)]^2} \]

For g(x) ≠ 0.

Chain Rule

\[ \frac{d}{dx}[f(g(x))] = f'(g(x)) \cdot g'(x) \]

This rule is essential for finding derivatives of composite functions.

Derivatives of Common Functions

Trigonometric Functions

\[ \frac{d}{dx}[\sin x] = \cos x \] \[ \frac{d}{dx}[\cos x] = -\sin x \] \[ \frac{d}{dx}[\tan x] = \sec^2 x \] \[ \frac{d}{dx}[\cot x] = -\csc^2 x \] \[ \frac{d}{dx}[\sec x] = \sec x \tan x \] \[ \frac{d}{dx}[\csc x] = -\csc x \cot x \]

Exponential and Logarithmic Functions

\[ \frac{d}{dx}[e^x] = e^x \] \[ \frac{d}{dx}[a^x] = a^x \ln a \] \[ \frac{d}{dx}[\ln x] = \frac{1}{x} \] \[ \frac{d}{dx}[\log_a x] = \frac{1}{x \ln a} \]

Inverse Trigonometric Functions

\[ \frac{d}{dx}[\arcsin x] = \frac{1}{\sqrt{1-x^2}} \] \[ \frac{d}{dx}[\arccos x] = -\frac{1}{\sqrt{1-x^2}} \] \[ \frac{d}{dx}[\arctan x] = \frac{1}{1+x^2} \]

Hyperbolic Functions

\[ \frac{d}{dx}[\sinh x] = \cosh x \] \[ \frac{d}{dx}[\cosh x] = \sinh x \] \[ \frac{d}{dx}[\tanh x] = \text{sech}^2 x \]

Higher-Order Derivatives

The derivative of a derivative is called the second derivative and is denoted by f''(x) or \(\frac{d^2y}{dx^2}\).

Notation

\[ f'(x) = \frac{df}{dx} = \frac{dy}{dx} \] \[ f''(x) = \frac{d^2f}{dx^2} = \frac{d^2y}{dx^2} \] \[ f^{(n)}(x) = \frac{d^nf}{dx^n} = \frac{d^ny}{dx^n} \]

The notation f^(n)(x) represents the nth derivative of f(x).

Physical Interpretation

In physics, higher-order derivatives often have specific physical meanings:

First derivative of position: velocity
Second derivative of position: acceleration
Third derivative of position: jerk
Fourth derivative of position: snap

These interpretations are crucial for understanding motion and dynamics.

Example: Motion Analysis

Consider a particle moving along a straight line with position function s(t) = t³ - 6t² + 9t + 1.

\[ s(t) = t^3 - 6t^2 + 9t + 1 \] \[ v(t) = s'(t) = 3t^2 - 12t + 9 \] \[ a(t) = v'(t) = s''(t) = 6t - 12 \] \[ j(t) = a'(t) = s'''(t) = 6 \]

At t = 2:

Position: s(2) = 8 - 24 + 18 + 1 = 3 units
Velocity: v(2) = 12 - 24 + 9 = -3 units/s (moving in negative direction)
Acceleration: a(2) = 12 - 12 = 0 units/s² (momentarily no acceleration)
Jerk: j(2) = 6 units/s³ (constant)

Partial Derivatives

When a function depends on multiple variables, we can find the rate of change with respect to each variable while holding the others constant.

Definition

For a function f(x, y), the partial derivatives are:

\[ \frac{\partial f}{\partial x} = \lim_{h \to 0} \frac{f(x + h, y) - f(x, y)}{h} \] \[ \frac{\partial f}{\partial y} = \lim_{h \to 0} \frac{f(x, y + h) - f(x, y)}{h} \]

These represent the rate of change of f with respect to one variable while holding the other(s) constant.

Notation

Common notations for partial derivatives include:

\[ \frac{\partial f}{\partial x}, \quad f_x, \quad D_x f \] \[ \frac{\partial f}{\partial y}, \quad f_y, \quad D_y f \]

Higher-order partial derivatives are denoted as:

\[ \frac{\partial^2 f}{\partial x^2}, \quad f_{xx} \] \[ \frac{\partial^2 f}{\partial y \partial x}, \quad f_{yx} \]

Example: Temperature Distribution

Consider a temperature distribution T(x, y) = x² + 2xy + y² on a metal plate.

\[ \frac{\partial T}{\partial x} = 2x + 2y \] \[ \frac{\partial T}{\partial y} = 2x + 2y \]

At the point (1, 2):

\[ \frac{\partial T}{\partial x}(1, 2) = 2(1) + 2(2) = 6 \] \[ \frac{\partial T}{\partial y}(1, 2) = 2(1) + 2(2) = 6 \]

This means that at the point (1, 2), the temperature increases at a rate of 6 degrees per unit distance in both the x and y directions.

Visualization of partial derivatives as slopes in different directions

Applications in Physics

Kinematics

Derivatives describe motion:

\[ \vec{v}(t) = \frac{d\vec{r}(t)}{dt} \] \[ \vec{a}(t) = \frac{d\vec{v}(t)}{dt} = \frac{d^2\vec{r}(t)}{dt^2} \]

These relationships allow us to analyze the motion of objects and predict their future positions.

Optimization

Critical points occur where the derivative equals zero:

\[ f'(x) = 0 \]

The second derivative test determines whether these points are maxima, minima, or neither:

\[ f''(x) < 0 \Rightarrow \text{maximum} \] \[ f''(x) > 0 \Rightarrow \text{minimum} \] \[ f''(x) = 0 \Rightarrow \text{inconclusive} \]

This is used to find optimal configurations in physical systems.

Wave Equations

The wave equation involves second derivatives:

\[ \frac{\partial^2 u}{\partial t^2} = c^2 \frac{\partial^2 u}{\partial x^2} \]

This equation describes the propagation of waves in various media, from sound waves to electromagnetic waves.

Thermodynamics

Partial derivatives define important thermodynamic quantities:

\[ C_V = \left(\frac{\partial U}{\partial T}\right)_V \] \[ \kappa_T = -\frac{1}{V}\left(\frac{\partial V}{\partial P}\right)_T \]

Where C_V is the heat capacity at constant volume and κ_T is the isothermal compressibility.

Example: Simple Harmonic Motion

A mass on a spring follows simple harmonic motion described by:

\[ x(t) = A\cos(\omega t + \phi) \]

The velocity and acceleration are:

\[ v(t) = \frac{dx}{dt} = -A\omega\sin(\omega t + \phi) \] \[ a(t) = \frac{dv}{dt} = -A\omega^2\cos(\omega t + \phi) = -\omega^2 x(t) \]

The last equation, a(t) = -ω²x(t), is the differential equation that defines simple harmonic motion. It states that the acceleration is proportional to the displacement but in the opposite direction, which is the hallmark of a restoring force.

Key Insight:

Derivatives are the mathematical language of change in physics. They allow us to describe how physical quantities evolve over time and space, formulate the fundamental laws of nature as differential equations, and analyze the behavior of complex systems. The ability to calculate and interpret derivatives is essential for understanding the dynamic nature of the physical world.

Practice Problems

Test your understanding of derivatives with these practice problems:

Find the derivative of f(x) = x³sin(x) using the product rule.
Calculate the second derivative of g(x) = e^(2x)ln(x).
Find the partial derivatives of f(x,y) = x²y + xy² + sin(xy).
A particle moves along a straight line with position function s(t) = t³ - 3t² + 2t. Find the times when the particle is at rest and determine whether it is changing direction at those times.
The temperature at a point (x,y) on a metal plate is given by T(x,y) = 100 - 2x² - y². Find the direction of maximum temperature decrease at the point (2,3).