1. Differential Calculus Core Formulae
Leibnitz’s Theorem for the \(n\)-th Derivative of a Product
Leibnitz Theorem
\[ (uv)_n = u_n v + \binom{n}{1} u_{n-1} v_1 + \binom{n}{2} u_{n-2} v_2 + \cdots + \binom{n}{r} u_{n-r} v_r + \cdots + u v_n \]
Crucial for finding the \(n\)-th derivative of expressions like \( x^2 e^{ax} \) or \( x^3 \sin x \).
Euler’s Theorem on Homogeneous Functions
If \( u = f(x, y) \) is a homogeneous function of degree \( n \) in \( x \) and \( y \), then:
Euler’s First & Second Orders
\[ x \frac{\partial u}{\partial x} + y \frac{\partial u}{\partial y} = n u \]
\[ x^2 \frac{\partial^2 u}{\partial x^2} + 2xy \frac{\partial^2 u}{\partial x \partial y} + y^2 \frac{\partial^2 u}{\partial y^2} = n(n-1) u \]
2. Integral Calculus: Beta & Gamma Functions
Definitions & Key Properties
Gamma Function (\( \Gamma(n) \))
\[ \Gamma(n) = \int_0^\infty e^{-x} x^{n-1} dx \quad (n > 0) \]
\[ \Gamma(n+1) = n \Gamma(n) = n! \quad (\text{for positive integers } n) \]
\[ \Gamma\left(\frac{1}{2}\right) = \sqrt{\pi} \]
Beta Function (\( B(m, n) \)) & Relation with Gamma
\[ B(m, n) = \int_0^1 x^{m-1} (1-x)^{n-1} dx = 2 \int_0^{\pi/2} \sin^{2m-1}\theta \cos^{2n-1}\theta \, d\theta \]
\[ B(m, n) = \frac{\Gamma(m) \Gamma(n)}{\Gamma(m+n)} \]
Wallis’s Definite Integral Formula
\[ \int_0^{\pi/2} \sin^p\theta \cos^q\theta \, d\theta = \frac{\Gamma\left(\frac{p+1}{2}\right) \Gamma\left(\frac{q+1}{2}\right)}{2 \Gamma\left(\frac{p+q+2}{2}\right)} \]
3. Matrices & Linear Algebra Shortcuts
- Trace of a Matrix: Sum of principal diagonal elements = Sum of Eigenvalues: \( \text{Trace}(A) = \sum \lambda_i \).
- Determinant: Product of all Eigenvalues = Determinant of matrix: \( \det(A) = \prod \lambda_i \).
- Eigenvalues of Inverse: If \( \lambda \) is an eigenvalue of non-singular matrix \( A \), then \( \frac{1}{\lambda} \) is an eigenvalue of \( A^{-1} \).
- Eigenvalues of Powers: If \( \lambda \) is an eigenvalue of \( A \), then \( \lambda^k \) is an eigenvalue of \( A^k \).
- Cayley-Hamilton Theorem: Every square matrix satisfies its own characteristic equation: \( |A - \lambda I| = 0 \).
4. Vector Calculus Integral Theorems
| Theorem | Mathematical Statement | Physical Domain |
|---|---|---|
| Green’s Theorem | \( \oint_C (M dx + N dy) = \iint_R \left(\frac{\partial N}{\partial x} - \frac{\partial M}{\partial y}\right) dx dy \) | Plane Curve to Region |
| Stokes’ Theorem | \( \oint_C \vec{F} \cdot d\vec{r} = \iint_S (\nabla \times \vec{F}) \cdot \hat{n} \, dS \) | Line Integral to Surface Integral |
| Gauss Divergence | \( \iint_S \vec{F} \cdot \hat{n} \, dS = \iiint_V (\nabla \cdot \vec{F}) \, dV \) | Closed Surface to Enclosed Volume |