Final hour summaries
Core Cheat Sheets
A one-page reference grid featuring key formulas, memory tricks, and quick-revision points for every subject.
A one-page reference grid featuring key formulas, memory tricks, and quick-revision points for every subject.
No grids, no metrics clutter, no widgets. Just pure consolidated academic facts and formula blocks.
Strings in C are simply contiguous arrays of characters backed by a Null-terminator character (\0). Memory locations computed by index displacements:
// Pointer arithmetic string loops
char *ptr = str;
while(*ptr != '\0') { ptr++; }
To calculate Matrix rank, transition array cells into Row Echelon reductions using simple subtraction operations:
1. Ensure maximum non-zero numbers shift upwards.
2. Count non-zero row maps after zeroes reach bottom limits.
3. Rank(A) = non-zero rows integer.
Every square matrix satisfies its own characteristic equation. If the characteristic equation is |A - λI| = 0, which expands to a polynomial p(λ) = λn + cn-1λn-1 + ... + c0 = 0, then by Cayley-Hamilton:
For a square matrix A, an eigenvalue λ and its corresponding eigenvector X satisfy the fundamental equation:
The fundamental equation of quantum mechanics describing stationary states of a particle of mass m in a potential field V:
Locates constrained optimization targets easily by scaling gradients matches:
Where λ (lambda) represents the scalar optimization coefficient matching constraint limits G.
Calculates diffraction grating slit displacement limits mathematically:
Calculates integrating factors for non-exact linear differential equations of format dy/dx + P(x)y = Q(x):
The composite solution reduces to:
Determine dynamic cell voltages or single electrode potentials across temperature and concentration gradients:
Under standard environmental fields (T = 298 K), the equation condenses elegantly to:
Establishes precise conditions where inductive reactances and capacitive reactances neutralize:
Solving for standard resonant frequency yields: