Mathematical demonstrations

Geometry of the triangle

Geometry of the circle

Trigonometry

Complex numbers

Properties of complex numbers\(: |z|, \ arg(z), \ \bar{z} \)
Moivre's formula\(: \Bigl[cos(\theta) + i \hspace{0.2em} sin(\theta)\Bigr]^n = cos(n\theta) + i \hspace{0.2em} sin(n\theta) \)

Geometry in space

Calculation of surfaces and volumes by integration\(: S_{sphere} = 4\pi R^2 \)
Properties of the scalar product\(: \vec{u}.\vec{v} = ||\vec{u}|| \times ||\vec{v}|| \times cos(\vec{u}, \vec{v})\)
Properties of the vector product\( : || \vec{u} \land \vec{v} || = || \vec{u}|| \times || \vec{v}|| \times sin(\vec{u}, \vec{v}) \)
Analytical geometry in space\(: M \in \mathcal{P}(A, \vec{n}) \hspace{0.1em} \Longleftrightarrow \hspace{0.1em} ax + by + cz + d = 0 \)

Polynomials

Limits and continuity

L'Hôpital's rule\(: \enspace \lim_{x \to \alpha} \enspace \frac{f(x)}{g(x)} = \lim_{x \to \alpha} \enspace \frac{f'(x)}{g'(x)} \)

Calculus

Integrals/antiderivatives

Differential equations

Asymptotic analysis

Taylor-Young's formula with integral remainder\(: f_{n,a}(x) = \hspace{0.2em} f(a) + f'(a)(x - a) \hspace{0.2em} + ...\hspace{0.2em} + \hspace{0.2em} {\displaystyle \int^x } \frac{f^{(n+1)}(t)}{n!}(x-t)^n \hspace{0.2em} dt \)

Miscellaneous

Logarithmic and exponential functions\(: ln(ab) = ln(a) + ln(b) \)

Mathematical demonstrations

Elementary calculation \( (a + b)^n = \sum \binom{n}{p} a^{n-p}b^p \)

Properties of fractions\(: \frac{a}{b} = \frac{c}{d} \Longleftrightarrow ad = bc \)

Properties of the powers of x (for natural exponents)\(: x^a x^b = x^{a+b}\)

Newton's binomial\(: (a + b)^n = \sum \binom{n}{p} a^{n-p}b^p \)

Geometrical identity\(: a^n - b^n = (a-b) \sum a^{n-p-1}b^p \)

Properties of the binom\(: \binom{n}{p} = \binom{n -1}{p -1 } + \binom{n - 1}{p} \)

Combinatorial analysis and enumerating formulas \( : \binom{n}{p} = \frac{n!}{p! (n-p)!} \)

Properties of matrix\(: (A \times B)_{i,j} = \sum a_{i,k} \times b_{k,j} \)

Geometry \( a \perp b \Longleftrightarrow a^2 + b^2 = c^2 \)

Pythagorean theorem and its reciprocal\(: a \perp b \Longleftrightarrow a^2 + b^2 = c^2 \)

Thales' theorem and its reciprocal\(: BC \parallel DE \Longleftrightarrow \frac{AB}{AD} = \frac{AC}{AE} = \frac{BC}{DE} \)

Geometric laws of a triangle\(: S_{abc} = \sqrt{p(p-a)(p-b)(p-c)} \)

Similarity of two triangles

Right-angled triangle inscribed in a circle

Calculation of Pi \( (\pi)\) by geometrical method

Potency of a point in relation to a circle\(: \overline{OA} \times \overline{OB} = \overline{OC} \times \overline{OD} \)

Trigonometric duplication and addition formulas\(: sin(2\alpha), \ sin(\alpha + \beta)... \)

Euler's trigonometric formulas\(: e^{ix} = cos(x) + i.sin(x) \)

Properties of complex numbers\(: |z|, \ arg(z), \ \bar{z} \)

Moivre's formula\(: \Bigl[cos(\theta) + i \hspace{0.2em} sin(\theta)\Bigr]^n = cos(n\theta) + i \hspace{0.2em} sin(n\theta) \)

Calculation of surfaces and volumes by integration\(: S_{sphere} = 4\pi R^2 \)

Properties of the scalar product\(: \vec{u}.\vec{v} = ||\vec{u}|| \times ||\vec{v}|| \times cos(\vec{u}, \vec{v})\)

Properties of the vector product\( : || \vec{u} \land \vec{v} || = || \vec{u}|| \times || \vec{v}|| \times sin(\vec{u}, \vec{v}) \)

Analytical geometry in space\(: M \in \mathcal{P}(A, \vec{n}) \hspace{0.1em} \Longleftrightarrow \hspace{0.1em} ax + by + cz + d = 0 \)

Analysis \( F(x) = F(a) + {\displaystyle \int_a^x } f(t) \ dt \)

Solving quadratic equations\(: P_2(X) = aX^2 + bX + c = 0 \)

Solving cubic equations\(: P_3(X) = aX^3 + bX^2 + cX + d = 0\)

Lagrange interpolating polynomial\(: L(X) = y_0 L_0(X) + y_1 L_1(X) \hspace{0.2em} + \hspace{0.2em} ... \hspace{0.2em} + \hspace{0.2em} y_n L_n(X) \)

L'Hôpital's rule\(: \enspace \lim_{x \to \alpha} \enspace \frac{f(x)}{g(x)} = \lim_{x \to \alpha} \enspace \frac{f'(x)}{g'(x)} \)

Derivativity\(: f'(x) = \lim_{\Delta_x \to 0} \ \frac{\Delta_{f(x)}}{\Delta_x} = \frac{df}{dx} \)

Derivatives of standard functions\(: (x^2)' , \ \big[ln(x)\big]' , \ \left(\frac{1}{x}\right)'... \)

Derivatives of trigonometric functions\(: sin(x)', \ cos(x)' , \ tan(x)'... \)

Derivatives of operations on functions\(: (f \circ g)' = g'(f' \circ g) \)

Convexity\(: f \enspace convex \enspace on \enspace I \Longleftrightarrow f(x) \geqslant f'(a)(x - a) + f(a) \)

Mean value theorem\(: \exists c \in \bigl ]a,b \bigr[, \ f'(c) = \frac{ f(b) - f(a)}{b-a} \)

Newton's method: a way to approximate any value

Length of a curve over an interval

Properties of integrals\(: {\displaystyle \int^x } \bigl(\lambda f + \mu g \bigr) \ dt = \lambda {\displaystyle \int^x } f \ dt + \mu {\displaystyle \int^x } g \ dt \)

Link between integrals and antiderivatives\(: F(x) = F(a) + {\displaystyle \int_a^x } f(t) \ dt \)

Standard antiderivatives and general integration methods\(: {\displaystyle \int_a^b } (f'g) \hspace{0.2em}dt = \Bigl[fg\Bigr]_{a}^b - {\displaystyle \int_a^b } (fg') \hspace{0.2em}dt \)

Integration methods for rational fractions \( : {\displaystyle \int^x } \frac{1}{t^2 + pt + q} \ dt = \frac{1}{(x_1 - x_2)} \ ln \left| \frac{x-x_1}{x-x_2} \right| \)

Integration methods for rational fractions with square roots \( : {\displaystyle \int^x } \frac{dt}{\sqrt{a^2 + t^2}} = ln \left|\sqrt{ a^2 + x^2 } + x\right| \)

Antiderivatives of trigonometric functions\(: {\displaystyle \int^x } tan(t) \ dt = - ln|cos(x)| \)

Solving 1st order linear differential equations with continuous coefficient\(: y' + a(x)y = f(x) \)

Solving 2nd order linear differential equations with constant coefficients\(: y'' + ay' + by= f(x) \)

Superposition principle

Taylor-Young's formula with integral remainder\(: f_{n,a}(x) = \hspace{0.2em} f(a) + f'(a)(x - a) \hspace{0.2em} + ...\hspace{0.2em} + \hspace{0.2em} {\displaystyle \int^x } \frac{f^{(n+1)}(t)}{n!}(x-t)^n \hspace{0.2em} dt \)

Logarithmic and exponential functions\(: ln(ab) = ln(a) + ln(b) \)

Sequences and series \( {\displaystyle \sum \bigl [a_{k+1} - a_k \bigr] = a_{n+1} - a_{0} } \)

Properties of numerical sequences\( : {\displaystyle \lim_{n \to \infty} \bigl[ u_n \bigr] = l \Longrightarrow \lim_{n \to \infty} \left[ \frac{1}{n} \sum_{k = 0}^n u_n \right] = l } \)

The Fibonacci sequence and the golden ratio\( : {\displaystyle F_n = \frac{\varphi^n}{\sqrt{5}} - \frac{(1 - \varphi)^n}{\sqrt{5}} } \)

Usual sums\(: \sum k, \ \sum k^2, \ \sum k^3, \ \sum (2k+1)... \)

Properties of sums\(: {\displaystyle \sum_{k=0}^n \bigl [a_{k+1} - a_k \bigr] = a_{n+1} - a_{0} } \)

Convergence criteria for series\(: {\displaystyle \int_{1}^{n + 1} f(t) \ dt \leqslant \sum_{k = 0}^n f(k) \leqslant \sum_{k = 1}^n \int_{0}^n f(t) \ dt } \)

Properties of convergent series\(: {\displaystyle \sum_{k = 0}^{+ \infty} (\lambda \ a_k) = \lambda \sum_{k = 0}^{+ \infty} a_k } \)

Arithmetic \( b \nmid a \Longrightarrow a = bq + R \)

Properties of divisibility\(: ka/kb \hspace{0.2em} \Longrightarrow \hspace{0.2em} a/b \)

Euclid's algorithm\(: b \nmid a \Longleftrightarrow a = bq + R \)

Properties of the GCD of two natural numbers\(: a = bq + R \Longrightarrow GCD(a, b) = GCD(b, R) \)

Properties of prime numbers\(: p \nmid a \hspace{0.2em} \Longleftrightarrow \hspace{0.2em} a \wedge p = 1 \)

Properties of congruences\(: a \equiv b \hspace{0.2em} \bigl[ n \bigr] \Longleftrightarrow n/(a -b) \)

Bezout's theorem ans its corollary\(: a \wedge b = 1 \Longleftrightarrow au + bv = 1 \)

Gauss' theorem ans its corollary\(: a / bc \enspace et \enspace a \wedge b = 1 \hspace{0.2em} \Longrightarrow \hspace{0.2em} a/c \)

Fermat's small theorem and its corollary\(: a^p \equiv a \hspace{0.2em} \bigl[p\bigr] \)

Glossary of symbols

Solving 1^st order linear differential equations with continuous coefficient\(: y' + a(x)y = f(x) \)

Solving 2^nd order linear differential equations with constant coefficients\(: y'' + ay' + by= f(x) \)