1. Determinants

A determinant is a scalar value that can be computed from the elements of a square matrix and encapsulates certain properties of the matrix.

• Definition: Determinants are associated with square matrices and are denoted as |A| for a matrix A.
• Minors: The determinant of a smaller matrix formed by deleting one row and one column of a matrix.
• Cofactors: Signed minors used in determinant expansions.
• Properties of Determinants:
  • Determinant of a triangular matrix is the product of its diagonal elements.
  • Swapping two rows or columns changes the sign of the determinant.
  • A determinant is zero if any two rows or columns are identical.

Example:
A = | 3  2 |
    | 5  4 |
Determinant, |A| = (3*4) - (2*5) = 2
        

2. Matrices

A matrix is a rectangular arrangement of numbers in rows and columns. It is a fundamental concept in linear algebra.

• Types of Matrices: Square Matrix, Diagonal Matrix, Identity Matrix, Zero Matrix.
• Operations:
  • Addition: Add corresponding elements of two matrices of the same dimension.
  • Subtraction: Subtract corresponding elements of two matrices of the same dimension.
  • Scalar Multiplication: Multiply each element of a matrix by a scalar value.
  • Matrix Multiplication: Multiply rows of the first matrix with columns of the second matrix.

Example:
A = [ [1, 2], [3, 4] ]
B = [ [5, 6], [7, 8] ]
Addition, A + B = [ [6, 8], [10, 12] ]
        

3. Adjoint and Inverse

The adjoint of a matrix is the transpose of its cofactor matrix, and the inverse is calculated using the formula:

A^-1 = adj(A)/|A|, where |A| ≠ 0.

Example:
A = [ [4, 7], [2, 6] ]
Adj(A) = [ [6, -7], [-2, 4] ]
|A| = (4*6) - (7*2) = 10
A-1 = 1/10 * [ [6, -7], [-2, 4] ]
        

4. Rank of a Matrix and Dependence of Vectors

The rank of a matrix is the maximum number of linearly independent rows or columns. The dependence of vectors occurs when a vector in a set can be expressed as a combination of other vectors.

Example:
A = [ [1, 2, 3], [4, 5, 6], [7, 8, 9] ]
After row reduction: Rank = 2
        

5. Eigenvalues and Eigenvectors

An eigenvector of a matrix A satisfies the equation A*v = λ*v, where λ is the eigenvalue.

Example:
A = [ [2, 1], [1, 2] ]
Eigenvalue λ = 3
Eigenvector v = [ [1], [1] ]
        

6. Cayley-Hamilton Theorem

The Cayley-Hamilton theorem states that every square matrix satisfies its own characteristic equation.

Example:
A = [ [1, 2], [3, 4] ]
Characteristic Equation: λ2 - 5λ + 2 = 0
A satisfies this equation.
        

1. Limit at a Point

The limit at a point describes the value that a function approaches as the input approaches a specific point.

• Mathematically, lim_x→c f(x) = L, means as x approaches c, the function f(x) approaches the value L.

Example:
f(x) = (x² - 1) / (x - 1)
limx→1 f(x) = limx→1 (x + 1) = 2
        

2. Properties of Limits

The following properties simplify the computation of limits:

• Sum Rule: lim_x→c [f(x) + g(x)] = lim_x→c f(x) + lim_x→c g(x)
• Product Rule: lim_x→c [f(x)·g(x)] = lim_x→c f(x) · lim_x→c g(x)
• Quotient Rule: lim_x→c [f(x)/g(x)] = lim_x→c f(x) / lim_x→c g(x), provided lim_x→c g(x) ≠ 0
• Constant Multiplication: lim_x→c [k·f(x)] = k · lim_x→c f(x)

Example:
f(x) = 2x, g(x) = x²
limx→2 [f(x) + g(x)] = limx→2 2x + limx→2 x² = 4 + 4 = 8
        

3. Computation of Limits of Various Types of Functions

Limits can be computed for different types of functions using direct substitution, factoring, rationalization, or L'Hôpital's Rule.

Example:
f(x) = (x² - 9) / (x - 3)
Factorize: (x - 3)(x + 3) / (x - 3)
limx→3 f(x) = 3 + 3 = 6
        

4. Continuity at a Point

A function is continuous at a point x = c if the following three conditions are satisfied:

• f(c) is defined.
• lim_x→c f(x) exists.
• lim_x→c f(x) = f(c).

Example:
f(x) = x²
At x = 2, f(2) = 4, limx→2 f(x) = 4. Hence, f(x) is continuous at x = 2.
        

5. Continuity Over an Interval

A function is continuous over an interval if it is continuous at every point in that interval.

Example:
f(x) = x² is continuous for all real numbers (-∞, ∞).
        

6. Intermediate Value Theorem

The Intermediate Value Theorem states that if a function is continuous on a closed interval [a, b], then for any value K between f(a) and f(b), there exists a value c in [a, b] such that f(c) = K.

Example:
f(x) = x³ - x
On [0, 2], f(0) = 0 and f(2) = 6. There exists a c ∈ (0, 2) such that f(c) = 3.
        

7. Types of Discontinuities

Discontinuities occur when a function is not continuous at a point. Types include:

• Removable Discontinuity: A "hole" in the graph where the limit exists but does not equal the function's value.
• Jump Discontinuity: The function "jumps" between two values at a point.
• Infinite Discontinuity: The function approaches infinity at a point.

Example:
f(x) = 1/(x - 1)
At x = 1, there is an infinite discontinuity.
        

1. Derivative

The derivative of a function represents the rate at which the function changes with respect to its variable.

Example:
f(x) = x²
f'(x) = 2x
        

2. Derivatives of Sum, Differences, Product & Quotients

• Sum Rule: (f + g)' = f' + g'
• Difference Rule: (f - g)' = f' - g'
• Product Rule: (fg)' = f'g + fg'
• Quotient Rule: (f/g)' = (f'g - fg')/g²

Example:
f(x) = x², g(x) = x + 1
(fg)' = (2x)(x + 1) + (x²)(1) = 3x² + 2x
        

3. Chain Rule

The chain rule is used to differentiate composite functions: (f(g(x)))' = f'(g(x)) · g'(x).

Example:
f(x) = (2x + 3)²
f'(x) = 2(2x + 3) · 2 = 4(2x + 3)
        

4. Derivatives of Composite Functions

Composite functions are differentiated using the chain rule.

Example:
f(x) = sin(2x)
f'(x) = cos(2x) · 2 = 2cos(2x)
        

5. Logarithmic Differentiation

Logarithmic differentiation simplifies differentiation of functions by taking the natural log of both sides.

Example:
f(x) = x^x
ln(f(x)) = xln(x)
f'(x) = x^x (1 + ln(x))
        

6. Rolle’s Theorem

Rolle's Theorem: If f(a) = f(b) and f(x) is continuous and differentiable on [a, b], then there exists a point c in (a, b) such that f'(c) = 0.

Example:
f(x) = x² - 4x + 4, on [0, 4]
f'(x) = 2x - 4
At x = 2, f'(x) = 0
        

7. Mean Value Theorem

Mean Value Theorem: If f(x) is continuous and differentiable on [a, b], then there exists c in (a, b) such that:

f'(c) = [f(b) - f(a)] / (b - a)

Example:
f(x) = x² on [1, 3]
f'(c) = (f(3) - f(1)) / (3 - 1) = 4
        

8. Expansion of Functions (Maclaurin's & Taylor's)

Maclaurin's Series: Expansion at x = 0.

Example:
f(x) = e^x
Maclaurin: 1 + x + x²/2! + x³/3! + ...
        

Taylor's Series: Expansion around x = a.

Example:
f(x) = ln(1 + x)
Taylor: x - x²/2 + x³/3 - ...
        

9. Indeterminate Forms & L’Hôpital’s Rule

L'Hôpital's Rule: For indeterminate forms (0/0 or ∞/∞), lim_x→c [f(x)/g(x)] = lim_x→c [f'(x)/g'(x)].

Example:
limx→0 (sin(x)/x) = limx→0 (cos(x)/1) = 1
        

10. Maxima & Minima

Maxima: Highest point of a function. Minima: Lowest point of a function.

Example:
f(x) = x² - 4x + 3
f'(x) = 2x - 4 = 0 at x = 2
Maxima/Minima: f''(2) > 0, so Minima at x = 2
        

11. Curve Tracing

Curve tracing involves studying a curve's behavior by analyzing derivatives and plotting its graph.

Example:
f(x) = x³ - 3x² + 2x
Critical points: x = 0, x = 1, x = 2
        

12. Successive Differentiation & Leibniz Theorem

Successive Differentiation: Repeatedly finding derivatives of a function.

Leibniz Theorem: Formula for nth derivative of a product of two functions.

Example:
f(x) = e^x sin(x)
fn(x) = e^x [sin(x + nπ/2)]
        

1. Integral as Limit of Sum

The integral as a limit of a sum defines the definite integral as the limit of Riemann sums, where a function is approximated by summing the areas of rectangles under its curve.

Example:
∫ f(x) dx from a to b = limn→∞ Σ [f(xi) Δx]
where Δx = (b - a)/n
        

2. Fundamental Theorem of Calculus (Without Proof)

The Fundamental Theorem of Calculus links differentiation and integration. It states:

• If f(x) is continuous on [a, b], and F(x) is its antiderivative, then:

∫ f(x) dx from a to b = F(b) - F(a)

Example:
f(x) = x²
∫ f(x) dx from 0 to 2 = F(2) - F(0) = (2³/3) - (0³/3) = 8/3
        

3. Indefinite Integrals

Indefinite integrals represent the family of all antiderivatives of a function.

Notation: ∫ f(x) dx = F(x) + C, where C is the constant of integration.

Example:
f(x) = 2x
∫ f(x) dx = x² + C
        

4. Methods of Integration

• Substitution Method: Replace a part of the integral with a substitution.

Example:
∫ sin(2x) dx
Let u = 2x, then du = 2 dx
∫ sin(u) du/2 = -cos(u)/2 + C = -cos(2x)/2 + C
            

• By Parts: ∫ u dv = uv - ∫ v du

Example:
∫ x e^x dx
Let u = x, dv = e^x dx
Then, ∫ x e^x dx = x e^x - ∫ e^x dx = x e^x - e^x + C
            

• Partial Fractions: Decompose a rational function into simpler fractions.

Example:
∫ (1 / (x² - 1)) dx
Decompose: 1 / (x² - 1) = 1/2(1/(x-1) - 1/(x+1))
Integrate: ∫ (1/2)(1/(x-1) - 1/(x+1)) dx = (1/2) ln|x-1| - (1/2) ln|x+1| + C
            

5. Reduction Formulae for Trigonometric Functions

Reduction formulae simplify the integration of higher powers of trigonometric functions.

Example:
∫ sinⁿ(x) dx = -(1/n) cos(x) sinⁿ⁻¹(x) + (n-1)/n ∫ sinⁿ⁻²(x) dx
        

6. Gamma and Beta Functions (Definition)

• Gamma Function: Defined as Γ(n) = ∫ xⁿ⁻¹ e⁻ˣ dx from 0 to ∞.

Example:
Γ(3) = ∫ x² e⁻ˣ dx from 0 to ∞ = 2
            

• Beta Function: Defined as B(x, y) = ∫ t^(x-1) (1-t)^(y-1) dt from 0 to 1.

Example:
B(1, 1) = ∫ t⁰ (1-t)⁰ dt from 0 to 1 = 1
            

1. Definition of a Vector in 2 and 3 Dimensions

A vector is a quantity that has both magnitude and direction. It is represented geometrically by an arrow pointing from one point to another.

• 2D Vector: Represented in the form v = (x, y), where x and y are components along the x-axis and y-axis.
• 3D Vector: Represented in the form v = (x, y, z), where x, y, and z are components along the x, y, and z axes respectively.

Example:
In 2D: v = (3, 4)
In 3D: v = (1, 2, 3)
Magnitude: |v| = √(x² + y²) in 2D, |v| = √(x² + y² + z²) in 3D
        

2. Double Scalar Product

The double scalar product of two vectors, also known as the dot product, is given by:

v · w = |v||w|cos(θ), where θ is the angle between the vectors.

Properties:

• v · w = w · v (Commutative)
• v · v = |v|²
• If v · w = 0, then v and w are perpendicular.

Example:
v = (3, 4), w = (1, 2)
v · w = (3)(1) + (4)(2) = 11
        

3. Triple Scalar Product

The triple scalar product of three vectors v, w, and u is given by:

v · (w × u), where w × u is the vector cross product.

This represents the volume of the parallelepiped formed by the three vectors.

Example:
v = (1, 0, 0), w = (0, 1, 0), u = (0, 0, 1)
w × u = (1, 0, 0)
v · (w × u) = 1 (Volume = 1 unit³)
        

4. Double Vector Product

The double vector product is given by:

(v × w) × u = (v · u)w - (w · u)v

This results in a vector that lies in the plane of w and u.

Example:
v = (1, 0, 0), w = (0, 1, 0), u = (0, 0, 1)
(v × w) × u = (0, 0, 1) × (0, 0, 1) = (0, 0, 0)
        

5. Physical Interpretation of Area and Volume

• Area: The magnitude of the cross product of two vectors represents the area of the parallelogram formed by the vectors.

Area = |v × w|

Example:
v = (1, 0, 0), w = (0, 1, 0)
v × w = (0, 0, 1)
Area = |(0, 0, 1)| = 1
            

• Volume: The magnitude of the triple scalar product represents the volume of the parallelepiped formed by three vectors.

Volume = |v · (w × u)|

Example:
v = (1, 0, 0), w = (0, 1, 0), u = (0, 0, 1)
Volume = |1| = 1