JEE Physics Mathematical Tools Notes Set 01

Functions and Angles

A function is a rule that gives exactly one value of a dependent variable \( y \) for each allowed value of the independent variable \( x \), written \( y=f(x) \). The set of allowed inputs is the domain and the set of outputs is the range. In physics almost every law is a function: position depends on time, force on extension, pressure on depth.
Angles are measured in degrees or in radians. One radian is the angle at the centre of a circle that cuts an arc equal to the radius, so \( \theta=\frac{\text{arc}}{\text{radius}} \) and \( \pi\ \text{rad}=180^\circ \). Always use radians in formulas such as \( s=r\theta \) and in calculus.

Trigonometry Essentials

In a right triangle: \( \sin\theta=\frac{\text{perpendicular}}{\text{hypotenuse}},\ \cos\theta=\frac{\text{base}}{\text{hypotenuse}},\ \tan\theta=\frac{\sin\theta}{\cos\theta} \). The reciprocals are \( \csc\theta,\ \sec\theta,\ \cot\theta \).
Identities worth remembering: \( \sin^2\theta+\cos^2\theta=1 \), \( 1+\tan^2\theta=\sec^2\theta \), \( \sin2\theta=2\sin\theta\cos\theta \), \( \cos2\theta=\cos^2\theta-\sin^2\theta=1-2\sin^2\theta \), \( \sin(A\pm B)=\sin A\cos B\pm\cos A\sin B \).
Standard values: \( \sin0^\circ=0,\ \sin30^\circ=\tfrac12,\ \sin45^\circ=\tfrac1{\sqrt2},\ \sin60^\circ=\tfrac{\sqrt3}2,\ \sin90^\circ=1 \); cosine runs through the same list in reverse. For small angles in radians, \( \sin\theta\approx\tan\theta\approx\theta \) and \( \cos\theta\approx1-\theta^2/2 \).
Signs follow the quadrant: all ratios are positive in the first, only sine in the second, only tangent in the third and only cosine in the fourth. Also \( \sin(180^\circ-\theta)=\sin\theta \), \( \cos(180^\circ-\theta)=-\cos\theta \), \( \sin(90^\circ-\theta)=\cos\theta \).

Differentiation

The derivative \( \frac{dy}{dx}=\lim_{\Delta x\to0}\frac{\Delta y}{\Delta x} \) is the instantaneous rate of change of \( y \) with \( x \), and geometrically the slope of the tangent to the graph. Velocity is \( \frac{dx}{dt} \) and acceleration is \( \frac{dv}{dt} \), so this single idea carries most of mechanics.
Basic results: \( \frac{d}{dx}x^n=nx^{n-1} \), \( \frac{d}{dx}\sin x=\cos x \), \( \frac{d}{dx}\cos x=-\sin x \), \( \frac{d}{dx}\tan x=\sec^2x \), \( \frac{d}{dx}e^x=e^x \), \( \frac{d}{dx}\ln x=\frac1x \), and the derivative of a constant is zero.

Rules of combination
1. Sum: \( (u\pm v)'=u'\pm v' \).
2. Product: \( (uv)'=u'v+uv' \).
3. Quotient: \( \left(\frac uv\right)'=\frac{u'v-uv'}{v^2} \).
4. Chain: \( \frac{dy}{dx}=\frac{dy}{du}\cdot\frac{du}{dx} \) for a function of a function, for example \( \frac{d}{dx}\sin(ax)=a\cos(ax) \).

Maxima, Minima and Rates

At a turning point of \( y=f(x) \) the slope is zero, \( \frac{dy}{dx}=0 \). If the second derivative \( \frac{d^2y}{dx^2} \) is negative there, the point is a maximum; if positive, a minimum. The second derivative tells how the slope itself is changing, which is why acceleration is the second derivative of position.
To solve an optimisation problem, write the quantity as a function of one variable, set its derivative to zero and check the sign of the second derivative. For related rates, relate the quantities by a formula and differentiate with respect to time, for example a rising water level in a vessel gives \( A\frac{dh}{dt}=\frac{dV}{dt} \).

Integration

Integration reverses differentiation: if \( \frac{dF}{dx}=f(x) \) then \( \int f(x)dx=F(x)+C \). The constant is fixed by an initial condition.
Basic results: \( \int x^ndx=\frac{x^{n+1}}{n+1}+C\ (n\neq-1) \), \( \int\frac{dx}{x}=\ln x+C \), \( \int\sin x\,dx=-\cos x+C \), \( \int\cos x\,dx=\sin x+C \), \( \int\sec^2x\,dx=\tan x+C \), \( \int e^xdx=e^x+C \), and \( \int\cos(ax)dx=\frac{\sin ax}{a}+C \).
For a composite integrand, substitute \( u=g(x) \) when \( g'(x) \) is present as a factor, then integrate in \( u \) and go back to \( x \).

A definite integral \( \int_a^bf(x)dx=F(b)-F(a) \) equals the signed area between the curve and the \( x \)-axis from \( a \) to \( b \). Area below the axis counts as negative, so split the range where the curve crosses zero when the actual area is wanted. In physics, the area under a velocity-time graph is displacement, under an acceleration-time graph is change in velocity, and under a force-displacement graph is work.

Scalars and Vectors

A scalar has magnitude only (mass, time, speed, energy); a vector has magnitude and direction and obeys the triangle or parallelogram law (displacement, velocity, force). Two vectors are equal only if both magnitude and direction match. A unit vector is \( \hat A=\vec A/|\vec A| \), and \( \hat i,\hat j,\hat k \) point along the three axes.
Addition: for vectors of magnitudes \( A \) and \( B \) with angle \( \theta \) between their directions, \( R=\sqrt{A^2+B^2+2AB\cos\theta} \) and the direction makes angle \( \alpha \) with \( \vec A \) where \( \tan\alpha=\frac{B\sin\theta}{A+B\cos\theta} \). The result ranges from \( |A-B| \) (opposite) to \( A+B \) (same direction). Subtraction is addition of the reversed vector: \( \vec A-\vec B=\vec A+(-\vec B) \), with magnitude \( \sqrt{A^2+B^2-2AB\cos\theta} \).

Resolution: a vector in a plane is \( \vec A=A_x\hat i+A_y\hat j \) with \( A_x=A\cos\phi \) and \( A_y=A\sin\phi \), where \( \phi \) is measured from the \( x \)-axis. To add many vectors, add all \( x \)-components and all \( y \)-components separately; the magnitude is \( \sqrt{R_x^2+R_y^2} \) and the angle is \( \tan^{-1}(R_y/R_x) \). A body in equilibrium has zero resultant, so both component sums are zero.

Dot Product and Cross Product

Scalar (dot) product: \( \vec A\cdot\vec B=AB\cos\theta=A_xB_x+A_yB_y+A_zB_z \). It is commutative, it is largest for parallel vectors, it vanishes for perpendicular vectors, and \( \vec A\cdot\vec A=A^2 \). Also \( \hat i\cdot\hat i=1 \) and \( \hat i\cdot\hat j=0 \). Work \( W=\vec F\cdot\vec s \) and power \( P=\vec F\cdot\vec v \) are dot products.
Vector (cross) product: \( \vec A\times\vec B=AB\sin\theta\,\hat n \) where \( \hat n \) is perpendicular to both, found by the right-hand rule. It is not commutative: \( \vec A\times\vec B=-\vec B\times\vec A \). It vanishes for parallel vectors and has magnitude equal to the area of the parallelogram on the two vectors. Also \( \hat i\times\hat j=\hat k,\ \hat j\times\hat k=\hat i,\ \hat k\times\hat i=\hat j \). Torque \( \vec\tau=\vec r\times\vec F \) is a cross product. In component form it is the determinant of the rows \( (\hat i,\hat j,\hat k),\ (A_x,A_y,A_z),\ (B_x,B_y,B_z) \).

Differentiating a Vector

If a vector changes with time, differentiate each component: \( \frac{d\vec A}{dt}=\frac{dA_x}{dt}\hat i+\frac{dA_y}{dt}\hat j+\frac{dA_z}{dt}\hat k \). Velocity is \( d\vec r/dt \) and acceleration is \( d\vec v/dt \). A vector of constant magnitude can still have a non-zero derivative if its direction changes, which is exactly what happens in circular motion. For a displacement through angle \( \theta \) on a circle of radius \( R \), the magnitude of the change in position vector is \( 2R\sin\frac\theta2 \).

Worked Examples

Example 1. Differentiate \( y=\sin^2x \). Using the chain rule with outer power 2: \( \frac{dy}{dx}=2\sin x\cos x=\sin2x \).

Example 2. Find the area under \( y=x^2 \) from \( x=0 \) to \( x=3 \). \( \int_0^3x^2dx=\frac{27}{3}=9 \) square units.

Example 3. Two forces of 6 N and 8 N act at right angles. \( R=\sqrt{36+64}=10 \) N, at \( \tan^{-1}(8/6)\approx53^\circ \) from the 6 N force.

Example 4. For \( \vec A=2\hat i+3\hat j \) and \( \vec B=\hat i-\hat j \): \( \vec A\cdot\vec B=2-3=-1 \) and \( \vec A\times\vec B=(2)(-1)-(3)(1)=-5 \) along \( \hat k \), that is \( -5\hat k \).

Example 5. A rectangle of fixed perimeter \( 2p \) has sides \( x \) and \( p-x \), area \( x(p-x) \). Setting \( p-2x=0 \) gives \( x=p/2 \); the second derivative is \( -2 \), so this is a maximum, a square.