@Mujtaba_Ali_
Solving the last problem The question is on surface integral using parameterised coordinates Looking at the cross product which means a perpendicular to the surface, we substitute the expression with r^2 sin phi Since the value of the function is not mentioned here, ill assume f(x,y,z) to be 1 Now: dS=r^2 sin phi d(theta)d(phi) Applying the given integrals both sides , On RHS, first doing theta: integral 0-2pi d theta = 2pi Integral 0 to pi ( sin phi)d phi= -cos phi and applying limits we get 2 Combining all expression we get r^2*2pi*2= 4pi r^2 which is area of a sphere
@Cr7-e7r1n
In India 1+1 =nursery 100÷5 = grade 1 (11x)+3= 36 is grade 6 Pythagoras theorem=7 Edit: Thanks for 10 k likes 👍
@caravenicetolosa8633
Arithmetic Operators
+ (addition)
- (subtraction)
× (multiplication)
÷ (division)
% (modulus)
^ (exponentiation)
Algebraic Symbols
= (equality)
≠ (inequality)
≈ (approximation)
∼ (similarity)
≃ (equivalence)
≤ (less than or equal to)
≥ (greater than or equal to)
< (less than)
> (greater than)
Geometric Symbols
∠ (angle)
Points and Lines
Point: A, B, C, ...
Line: l, m, n, ...
Ray: → (e.g., AB→)
Line Segment: ‾‾ (e.g., AB‾‾)
Angles
Angle: ∠ (e.g., ∠ABC)
Right Angle: ⊥ (e.g., ∠ABC ⊥)
Straight Angle: ⌒ (e.g., ∠ABC ⌒)
Shapes
Triangle: Δ (e.g., ΔABC)
Quadrilateral: □ (e.g., ABCD□)
Pentagon: ⏀ (e.g., ABCDE⏀)
Hexagon: ⏁ (e.g., ABCDEF⏁)
Circle: ○ (e.g., O○)
Measurements
Congruent: ≅ (e.g., AB ≅ CD)
Similar: ∼ (e.g., ΔABC ∼ ΔDEF)
Perpendicular: ⊥ (e.g., AB ⊥ CD)
Parallel: ‖ (e.g., AB ‖ CD)
Operations
Intersection: ∩ (e.g., AB ∩ CD)
Union: ∪ (e.g., AB ∪ CD)
Complement: ′ (e.g., AB′)
Miscellaneous
Diameter: d (e.g., d = AB)
Radius: r (e.g., r = OA)
Circumference: C (e.g., C = 2πr)
Area: A (e.g., A = πr²)
∟ (right angle)
⊥ (perpendicular)
∥ (parallel)
~(similarity)
≅ (congruence)
Sine - sin
Cosine - cos
Tangent - tan
Cotangent - cot
Secant - sec
Cosecant - csc
Inverse Sine - sin^-1 or arcsin
Inverse Cosine - cos^-1 or arccos
Inverse Tangent - tan^-1 or arctan
Inverse Cotangent - cot^-1 or arccot
Inverse Secant - sec^-1 or arcsec
Inverse Cosecant - csc^-1 or arccsc
Here are the main trigonometry formulas:
Basic Trigonometry Formulas
1. sin(θ) = opposite side / hypotenuse
2. cos(θ) = adjacent side / hypotenuse
3. tan(θ) = opposite side / adjacent side
4. cot(θ) = adjacent side / opposite side
5. sec(θ) = hypotenuse / adjacent side
6. csc(θ) = hypotenuse / opposite side
Pythagorean Identities
1. sin²(θ) + cos²(θ) = 1
2. tan²(θ) + 1 = sec²(θ)
3. cot²(θ) + 1 = csc²(θ)
Sum and Difference Formulas
1. sin(a + b) = sin(a)cos(b) + cos(a)sin(b)
2. sin(a - b) = sin(a)cos(b) - cos(a)sin(b)
3. cos(a + b) = cos(a)cos(b) - sin(a)sin(b)
4. cos(a - b) = cos(a)cos(b) + sin(a)sin(b)
5. tan(a + b) = (tan(a) + tan(b)) / (1 - tan(a)tan(b))
6. tan(a - b) = (tan(a) - tan(b)) / (1 + tan(a)tan(b))
Double Angle Formulas
1. sin(2a) = 2sin(a)cos(a)
2. cos(2a) = cos²(a) - sin²(a)
3. tan(2a) = 2tan(a) / (1 - tan²(a))
Half Angle Formulas
1. sin(a/2) = ±√((1 - cos(a)) / 2)
2. cos(a/2) = ±√((1 + cos(a)) / 2)
3. tan(a/2) = ±√((1 - cos(a)) / (1 + cos(a)))
Calculus Symbols
∫ (integral)
∂ (partial derivative)
∇ (gradient)
Δ (delta)
∞ (infinity)
Logical Symbols
∧ (conjunction)
∨ (disjunction)
¬ (negation)
⇒ (implication)
⇔ (equivalence)
Set Theory Symbols
∈ (element of)
∉ (not an element of)
⊆ (subset)
⊇ (superset)
∪ (union)
∩ (intersection)
∅ (empty set)
Miscellaneous Symbols
! (factorial)
√ (square root)
| | (absolute value)
≈ (approximation)
∝ (proportionality)
Greek Letters
α (alpha)
β (beta)
γ (gamma)
δ (delta)
ε (epsilon)
ζ (zeta)
η (eta)
θ (theta)
ι (iota)
κ (kappa)
λ (lambda)
μ (mu)
ν (nu)
ξ (xi)
ο (omicron)
π (pi)
ρ (rho)
σ (sigma)
τ (tau)
υ (upsilon)
φ (phi)
χ (chi)
ψ (psi)
ω (omega)
Mathematical Constants
e (Euler's number)
i (imaginary unit)
π (pi)
τ (tau)
Operators
∑ (summation)
∏ (product)
∫ (integral)
∮ (contour integral)
∴ (therefore)
∵ (because)
Relations
≡ (equivalence relation)
≈ (approximation)
∼ (similarity)
≃ (equivalence)
≤ (less than or equal to)
≥ (greater than or equal to)
< (less than)
> (greater than)
= (equality)
≠ (inequality)
Other Symbols
⁰ (zero superscript)
¹ (one superscript)
² (squared)
³ (cubed)
ⁿ (n superscript)
₊ (plus sign in superscript)
₋ (minus sign in superscript)
₌ (approaches the limit)
₍₎ (floor function)
₎ (ceiling functional)
Formulas of :
Algebra
1. Linear Equations: ax + by = c
2. Quadratic Formula: x = (-b ± √(b² - 4ac)) / 2a
3. Polynomial Factorization: a(x - r₁)(x - r₂)...(x - rₙ)
4. Binomial Theorem: (a + b)ⁿ = ∑[n! / (k!(n-k)!)] * a^(n-k) * b^k
Calculus
1. Limit Definition: lim x→a f(x) = L
2. Derivative: f'(x) = lim h→0 [f(x + h) - f(x)] / h
3. Integral: ∫f(x) dx = F(x) + C
4. Fundamental Theorem of Calculus: ∫f(x) dx = F(b) - F(a)
Geometry
1. Pythagorean Theorem: a² + b² = c²
2. Circle Area: A = πr²
3. Sphere Volume: V = (4/3)πr³
4. Triangle Area: A = (1/2)bh
Statistics and Probability
1. Mean: μ = (Σxᵢ) / n
2. Median: M = (n+1)/2 th term
3. Standard Deviation: σ = √(Σ(xᵢ - μ)² / n)
4. Probability: P(A) = (Number of favorable outcomes) / (Total number of outcomes)
Number Theory
1. Divisibility Rules: (e.g., 2, 3, 4, 5, ...)
2. Prime Factorization: n = p₁^a¹ * p₂^a² * ... * pₖ^aₖ
3. Greatest Common Divisor (GCD): gcd(a, b) = max{x | x divides a and x divides b}
4. Fermat's Little Theorem: a^(p-1) ≡ 1 (mod p)
@paminipremalal1568
This is the real truth that adults cannot understand at all. 😢
@DaMostAvrageSoilders
and job is: can of beans + X5 soda crates + food + decoration + litterally toys. + working = cash edit o_o what in the world 4.5K LIKES?! PLUS A HEART?!?!?!?!?!? CRAZYYYY Other edit: ok nvm it got removed
@eddiechew5249
First integrate using the surface area (dx/dt)^2+(dy/dt)^2. Looking at the cross product so we let S= r^2sin phi be the perpendicular to the surface. No function is given, we cannot assume that the function is 1 Integrate S, dS: integral of r^2sin phi d(theta) with limits 0 to 2pi, which you will get 2pi. Then you integrate the function of sinphi d(phi) with limits 0 to pi, you will get 2. When you integrate with sinphi d(phi), you must get - cos phi. Hence, we get the expression of r^2(2pi)^2, and it will be 4(pi)r^2. Therefore this is the area of a sphere
@universalC2
maths goes from numbers to letters to Egyptian hieroglyphics
@Not_Hanz
That slap killed me 😭✋️
@eternity_12
For India (in my school) Nursery : 1+1 kindergarten: table of 2 1st grade: multiplication, division, addition and subtraction 3rd grade: DMAS 4th grade: BODMAS 5th grade: algebra 6th grade: linear equations 7th grade: Pythagoras theorem and trigonometry 8th: distance formula and shit Then it’s the same
@Hamdan4k
This what sin cosin tan is: Sine (sin), cosine (cos), and tangent (tan) are basic trigonometric functions that describe the relationship between the angles and sides of a right triangle. In a Right Triangle: For an angle θ in a right triangle: sin(θ) = opposite / hypotenuse cos(θ) = adjacent / hypotenuse tan(θ) = opposite / adjacent On the Unit Circle: sin(θ) is the y-coordinate. cos(θ) is the x-coordinate. tan(θ) = sin(θ) / cos(θ) = y / x (except where cos(θ) = 0).
@HarshHingane
We have trigonometry in class 10☠️ And we learnt pythagorean theorem in class 7 Here in india
@printerhellow
The slap was personal 💀
@MasterMalayali
I miss the day when maths was just numbers
@Lindsey8667
America (my school) Kindergarten: Count from 1-100 and add using tables 1st Grade: 1+1 and more basic addition 2nd Grade: Harder Addition and Subtraction such as 292-163 and multiplication from 1-6 3rd Grade: Times tables 1-12 4th Grade: Division and Fractions 5th Grade: Pemdas 7 (3+8) 6th Grade: Lcm and Gcf and complex basic math 7th Grade: Equations 3x+8=29 8th Grade: Pre-Algebra 9th Grade: Algebra 10th Grade: Geometry 11th Grade: Trigonometry 12th Grade: Calculus +1 12th Grade: College I +2 12th Grade: College II
@Queen_Of_Editsss
(11x) + 3 = 36 11x = 36 - 3 = 33 x = 33 ÷ 11 = 3 x = 3
@SheBelievedHim
The formulas: Formula of P.G.T = H²=B²+P² H² = Hypotenuse² B² = Base² P² = Perpendicular² Perpendicular² is also known as altitude² Hypotenuse is the longest side in the right angle Base is the __ part in the right angle Perpendicular is the straight part of the right angle P.G.T= Pythagoras Theorem Solving the 3rd problem (11x)+3=36 11x=36-3 [we reverse the symbols for example: +=-,-=+] 11x=33 x=33 (divided by) 11 =3 Edit: if u understand the formulas, u will find it easy as 1st grade, I hope this helps. Edit 2: TYSM FOR OVER 100 LIKES GUYS!!
@purpledangerousonic269
In my school: Nursery: simple addition and subtraction, 1+1 2-1 KG: Multiplication Tables and division readiness 1: Simple division 2: same 3: multiplication with 2 digit and 2 digit division 4: 3 digit division and multiplication factor and multiples lcm hcf 5: ?
@mobil3toxin569
Let x = null, y = null; let cosx, cosy; let angle = math.atan2(x-cosy, y-cosx) return Math.cos(cosx) * 80, Math.sin(cosy) * 80; u have 20 minutes to answer what direction x and y would be facing and the radius and the angle
@AmNotNuclear
Actual levels: Year 1: Number bonds and simple addition/subtraction Year 3: Introduction of multiplication and division Year 5/6: Basic geometry and other smaller topics Year 7: Pythagoras theorem (how do people find it hard?) Year 9: Sin, tan, and cosine Year 10: Quadratic formula and functions
@Navi-g5z