Limits of function of two variables

Limits of Functions of Two Variables

Definition

Let f(x,y) be defined in a neighborhood of the point (a,b), except possibly at the point itself. We say the limit of f(x,y) as (x,y) → (a,b) is L, and write:

lim_{(x,y)→(a,b)} f(x,y) = L

if for every ε>0, there exists δ>0 such that:

0 < √((x-a)² + (y-b)²) < δ ⇒ |f(x,y)-L| < ε

Intuitive Explanation

The value of f(x,y) gets closer to L as (x,y) approaches (a,b) from any direction.
The limit must be the same along all paths approaching (a,b).

Examples

Example 1

Function: f(x,y) = x + y

Find: lim_{(x,y)→(1,2)} f(x,y)

Solution: f(1,2) = 1 + 2 = 3

Answer: 3

Example 2

Function: f(x,y) = xy / (x² + y²)

Find: lim_{(x,y)→(0,0)} f(x,y)

Solution:

Along y=0: f(x,0)=0
Along x=0: f(0,y)=0
Along y=x: f(x,x)=1/2

Answer: Limit does not exist (different values along different paths)

Example 3

Function: f(x,y) = x² + y²

Find: lim_{(x,y)→(1,1)} f(x,y)

Solution: Direct substitution: f(1,1) = 1² + 1² = 2

Answer: 2

Example 4

Function: f(x,y) = (x² - y²) / (x² + y²)

Find: lim_{(x,y)→(0,0)} f(x,y)

Solution:

Along y=0: f(x,0)=1
Along x=0: f(0,y)=-1

Answer: Limit does not exist

Example 5

Function: f(x,y) = 3x²y / (x² + y²)

Find: lim_{(x,y)→(0,0)} f(x,y)

Solution:

Along y=0: f(x,0)=0
Along x=0: f(0,y)=0
Along y=kx: f(x,kx) → 0 as x→0

Answer: 0

Summary Table

Function	Point	Limit	Exists?
x + y	(1,2)	3	✔
xy / (x² + y²)	(0,0)	–	✘
x² + y²	(1,1)	2	✔
(x² - y²) / (x² + y²)	(0,0)	–	✘
3x²y / (x² + y²)	(0,0)	0	✔

Function of Two variable:

Function of Two variable: Let u be a symbol which has a definite value for every pair of values of x and y, then u is called a function of two independent variable x and y and is written as

u=f(x,y)

Function of Two Variables – Graphical Representation

A function of two variables is written as:

z = f(x, y)

1. General 3D Graph

          z
          |
          |        •
          |      •
          |    •
          |  •
          |•____________ y
         /
        /
       x

Explanation: This shows a surface in 3D space where z depends on x and y.

2. Plane Surface (z = x + y)

          z
          |
          |      /
          |    /
          |  /
          |/________ y
         /
        /
       x

Application: Cost, temperature variation.

3. Paraboloid (z = x² + y²)

          z
          |
        __|__
      /   |   \
    /     |     \
  /_______|_______\ y
          |
          x

Application: Heat distribution, potential energy.

4. Saddle Surface (z = xy)

          z
          |
      ___/ \___
     /           \
----/-------------\---- y
    \             /
      \___   ___/
            x

Application: Profit–loss analysis.

5. Contour Diagram (x² + y² = c)

        y
        |
     ○  ○  ○
   ○     ○
 ○    ○    ○
   ○     ○
     ○  ○
        |
        x

Application: Topographic maps, weather maps.

Exam Note:
The graph of a function of two variables is a surface in three-dimensional space.

Function of Three Variables

A function of three variables is a function that depends on three independent variables.

Mathematical Form:

w = f(x, y, z)

Examples of Functions of Three Variables

Linear: f(x, y, z) = x + y + z
Quadratic: f(x, y, z) = x² + y² + z²
Product: f(x, y, z) = xyz
Trigonometric: f(x, y, z) = sin x + cos y + tan z
Rational: f(x, y, z) = (x + y) / z, z ≠ 0

Graphical Representation

A function of three variables cannot be drawn directly because it requires four dimensions. Hence, it is represented using level surfaces or cross-sections.

Level Surface Example

For: x² + y² + z² = c (Sphere)

          z
          |
       ___|___
    .-'    |    '-.
  .'        |        '.
 |          |          |
  '.        |        .'
    '-._____|_____.-'
          |
          x
         /
        y

Explanation: Each surface represents points where the function has the same value.

Cross-Section Concept

Fix one variable (e.g., z = k) and draw the 3D surface in x–y plane.

        z = k
          |
      ____|____
     /    |    \
    |     |     |
     \____|____/
          |
          x
         /
        y

Real-Life Applications

Temperature in a room: T = f(x, y, z)
Pressure in fluids
Electric and gravitational potential
Air pollution concentration
Medical imaging (CT, MRI)

Exam Note:
A function of three variables is represented graphically using level surfaces or cross-sections.

Assignment Functions of Two and Three Variables

Functions of Two and Three Variables

Section 1: Functions of Two Variables

Definition

A function of two variables depends on two independent variables x and y and produces a single dependent variable z:

z = f(x, y)

Examples

f(x, y) = x + y
f(x, y) = x² + y²
f(x, y) = xy
f(x, y) = sin x + cos y
f(x, y) = x / y, y ≠ 0

Graphical Representation

Graph is a surface in 3D space.
Plane: z = x + y
Paraboloid: z = x² + y²
Saddle: z = xy

Level Curves

A level curve is given by f(x,y) = c, e.g., x² + y² = 1.

Applications

Temperature distribution
Population density
Cost analysis
Pressure on surfaces
Topography

Section 2: Functions of Three Variables

Definition

A function of three variables depends on x, y, z and produces a single output w:

w = f(x, y, z)

Examples

f(x,y,z) = x + y + z
f(x,y,z) = xyz
f(x,y,z) = x² + y² + z²
f(x,y,z) = sin x + cos y + tan z
f(x,y,z) = (x+y)/z, z ≠ 0

Graphical Representation

Cannot be plotted directly in 3D (requires 4D).
Use level surfaces: f(x,y,z) = c
Use cross-sections: fix one variable and plot remaining 2D function

Applications

Temperature in a room
Pressure in fluids
Air pollution modeling
Electric/gravitational fields
Medical imaging (CT/MRI)

Section 3: Questions and Answers

Functions of Two Variables

Q1: Define a function of two variables.

A1: A function of two variables depends on x and y and produces a single output z.

Q2: Write the general form of a function of two variables.

A2: z = f(x, y)

Q3: How many dimensions are needed to graph it?

A3: Three dimensions (x, y, z)

Q4: What is a level curve? Give an example.

A4: A curve where f(x,y) = c, e.g., x² + y² = 1

Q5: Give any two examples.

A5: f(x,y)=x+y, f(x,y)=x²+y²

Q6: If f(x,y)=x²+y², find f(2,3).

A6: f(2,3)=2²+3²=13

Q7: Find f(-2,4) for f(x,y)=xy.

A7: f(-2,4)=(-2)(4)=-8

Q8: Determine domain of f(x,y)=x/y.

A8: All real x,y with y≠0

Functions of Three Variables

Q1: Define a function of three variables.

A1: Depends on x, y, z and produces a single output w.

Q2: Write the general form.

A2: w = f(x, y, z)

Q3: Why cannot it be plotted directly?

A3: Requires 4D visualization; use level surfaces or cross-sections.

Q4: Give three examples.

A4: f(x,y,z) = x+y+z, f(x,y,z) = xyz, f(x,y,z)=x²+y²+z²

Q5: What is a level surface? Give example.

A5: Surface where f(x,y,z)=c, e.g., x²+y²+z²=1 (sphere)

Q6: If f(x,y,z)=x+y+z, find f(1,2,3).

A6: 6

Q7: If f(x,y,z)=xyz, find f(2,-1,3).

A7: -6

Q8: Determine domain of f(x,y,z)=(x+y)/z.

A8: All real x,y,z with z≠0

Section 4: Diagrams

General Graph of Function of Two Variables z | | • | • | • | • |•____________ y / / x

Plane Surface (z = x + y) z | | / | / | / |/________ y / / x

Paraboloid (z = x² + y²) z | __|__ / | \ / | \ /_______|_______\ y | x

Saddle Surface (z = xy) z | ___/ \___ / \ /-------------\---- y \ / \___ ___/ x

Level Curve (x² + y² = c) y | ○ ○ ○ ○ ○ ○ ○ ○ ○ ○ ○ ○ | x

Level Surface for Function of Three Variables (x² + y² + z² = c) z | ___|___ .-' | '-. .' | '. | | | '. | .' '-.__|__. -' | x / y

Cross-Section for Function of Three Variables z = k | ____|____ / | \ | | | \____|____/ | x / y

Calculus of Several Variables

Table of Content

1	Functions of two variables
2	Limits of function of two variables
3	Continuity of function of two variables
4	Partial derivatives
5	Partial derivatives
6	Partial derivatives
7	Total derivatives
8	Total derivatives
9	Maxima and minima
. 10	Lagrange multipliers method.

Statistical and Computational Optimization

Unit -1

Big Data Matrix

What is a Big Data Matrix?

A Big Data Matrix is a very large data structure arranged in rows and columns, where:

Rows represent entities (users, sensors, documents)
Columns represent features or attributes
Data volume is too large for traditional systems

Characteristics of Big Data Matrix

Large scale (millions or billions of rows)
High dimensionality
Sparse data (many zero values)
Stored and processed in distributed systems

Why Big Data Matrices are Important

Used in Machine Learning and Artificial Intelligence
Helps in data analytics and decision making
Supports large-scale scientific research
Enables real-time data processing

Types of Data Matrix

Numerical Data Matrix
Binary Data Matrix
Categorical Data Matrix
Sparse Data Matrix
Distributed Data Matrix

1. Numerical Data Matrix

Definition

A Numerical Data Matrix is a matrix in which all elements are numerical values.

$$ X = \begin{bmatrix} x_{11} & x_{12} & \cdots & x_{1n} \\ x_{21} & x_{22} & \cdots & x_{2n} \\ \vdots & \vdots & \ddots & \vdots \\ x_{m1} & x_{m2} & \cdots & x_{mn} \end{bmatrix} $$

Rows represent observations
Columns represent variables
$x_{ij} \in \mathbb{R}$

Example 1: Student Marks Matrix

$$ M = \begin{bmatrix} 78 & 85 & 90 \\ 88 & 76 & 84 \\ 92 & 89 & 95 \end{bmatrix} $$

Average Marks of Student i:

$$ \mu_i = \frac{1}{n}\sum_{j=1}^{n} M_{ij} $$

Example 2: Weather Observation Matrix

$$ W = \begin{bmatrix} 30.5 & 78 & 12 \\ 29.8 & 82 & 5 \\ 31.2 & 75 & 0 \end{bmatrix} $$

Mean Temperature:

$$ \bar{T} = \frac{1}{m}\sum_{i=1}^{m} W_{i1} $$

Example 3: Sales Data Matrix

$$ S = \begin{bmatrix} 1200 & 1350 & 1100 \\ 900 & 1000 & 980 \\ 1500 & 1600 & 1700 \end{bmatrix} $$

Total Sales:

$$ \text{Total} = \sum_{i=1}^{m}\sum_{j=1}^{n} S_{ij} $$

Example 4: Sensor Data Matrix

$$ R = \begin{bmatrix} 22.4 & 101.3 \\ 22.6 & 101.1 \\ 22.5 & 101.2 \end{bmatrix} $$

Variance of Temperature:

$$ \sigma^2 = \frac{1}{m}\sum_{i=1}^{m}(R_{i1}-\mu)^2 $$

Example 5: Image Pixel Matrix

$$ I = \begin{bmatrix} 0 & 128 & 255 \\ 64 & 192 & 128 \\ 255 & 128 & 0 \end{bmatrix} $$

Normalization:

$$ I' = \frac{I}{255} $$

Big Data Matrix – Real World Examples

1. User–Item Matrix

Rows: Users
Columns: Products
Used in Amazon, Netflix, Flipkart

2. Social Network Matrix

Rows and Columns: Users
Values: 1 (connected), 0 (not connected)

3. Sensor Data Matrix

Rows: Time intervals
Columns: Sensors

4. Term–Document Matrix

Rows: Documents
Columns: Words

5. Healthcare Data Matrix

Rows: Patients
Columns: Medical attributes

Summary

Big Data Matrices represent massive real-world datasets
They are sparse, high-dimensional, and distributed
Widely used in AI, analytics, healthcare, and IoT

Thank You

Integration by Partial Fraction

📥 Download Partial Fractions PDF

Integration by Partial Fractions — Examples & Solutions

Integration by Partial Fractions

Types and worked examples (converted from the uploaded notes).

Type 1: Distinct linear factors

When the denominator is expressible as a product of distinct linear factors.

Example 1

Evaluate $ \displaystyle \int \frac{x-1}{(x+1)(x-2)} \, dx $.

Write

\( \dfrac{x-1}{(x+1)(x-2)} = \dfrac{A}{x+1} + \dfrac{B}{x-2} \)

Then

\( x-1 = A(x-2) + B(x+1) \)

Putting $x=2$: $1 = 3B \Rightarrow B=\tfrac{1}{3}$.
Putting $x=-1$: $-2 = -3A \Rightarrow A=\tfrac{2}{3}$.

Thus

\( \dfrac{x-1}{(x+1)(x-2)} = \dfrac{2/3}{x+1} + \dfrac{1/3}{x-2} \)

Integrate:

\( \int \frac{x-1}{(x+1)(x-2)} dx = \frac{2}{3}\ln|x+1| + \frac{1}{3}\ln|x-2| + C \)

Example 2

Evaluate $ \displaystyle \int \frac{2x-1}{(x-1)(x+2)(x-3)} \, dx $.

Assume

\( \dfrac{2x-1}{(x-1)(x+2)(x-3)} = \dfrac{A}{x-1} + \dfrac{B}{x+2} + \dfrac{C}{x-3} \)

Multiply through and compare or use substitution at roots.

Putting $x=-2$: $-5 = ( -3)(-5)B \Rightarrow B=-\tfrac{1}{3}$.
Putting $x=1$: $1 = (3)(-2)A \Rightarrow A=-\tfrac{1}{6}$.
Putting $x=3$: $5 = 2\cdot5\;C \Rightarrow C=\tfrac{1}{2}$.

\( \dfrac{2x-1}{(x-1)(x+2)(x-3)} 
= -\tfrac{1}{6}\dfrac{1}{x-1} -\tfrac{1}{3}\dfrac{1}{x+2} + \tfrac{1}{2}\dfrac{1}{x-3}
\)

Integrate:

\( \int \dfrac{2x-1}{(x-1)(x+2)(x-3)} dx 
= -\tfrac{1}{6}\ln|x-1| -\tfrac{1}{3}\ln|x+2| + \tfrac{1}{2}\ln|x-3| + C
\)

Type II: Repeated linear factors

When denominator contains repeating linear factors.

Example 3

Evaluate $ \displaystyle \int \frac{3x+1}{(x-2)^2(x+2)} \, dx $.

Assume decomposition of the form

\( \dfrac{3x+1}{(x-2)^2(x+2)} = \dfrac{A}{x-2} + \dfrac{B}{(x-2)^2} + \dfrac{C}{x+2} \)

Solving (substitute roots & compare coefficients) gives $B=\tfrac{4}{7}$, $C=-\tfrac{5}{16}$, and $A=\tfrac{5}{16}$.

Thus integrate to get logarithms and simple rational terms. (Detailed algebra steps in original notes.)

Example 4

This example in the notes follows the same repeated-factor method — substitute roots and compare coefficients to find constants, then integrate term-by-term to obtain log terms and rational terms.

Type III: Irreducible quadratic factors

When denominator has irreducible quadratic factors, use linear numerators for those parts.

Example 5

Evaluate $ \displaystyle \int \frac{8}{(x+2)(x^2+4)} \, dx $.

Decompose as

\( \dfrac{8}{(x+2)(x^2+4)} = \dfrac{A}{x+2} + \dfrac{Bx+C}{x^2+4} \)

Solving constants yields $A=1$, $B=-1$, $C=2$.

\( \dfrac{8}{(x+2)(x^2+4)} = \dfrac{1}{x+2} + \dfrac{-x+2}{x^2+4} \)
\end{pre>
        Integration gives
        \( \int \dfrac{8}{(x+2)(x^2+4)} dx 
= \ln|x+2| - \tfrac{1}{2}\ln(x^2+4) + \arctan\left(\tfrac{x}{2}\right) + C
\)

Types of Matrix

Types of Matrices — Definitions & Examples

A concise reference listing common matrix types with a short definition and two examples for each. Perfect for students, teachers, and blog readers.

Row Matrix

Definition: A matrix with only one row.

Examples

[ 3  5  7 ]
[ -2  4  9  1 ]

Column Matrix

Definition: A matrix with only one column.

Examples

[ 4
 -1
  2 ]

[ 7
  0 ]

Square Matrix

Definition: A matrix with an equal number of rows and columns (n × n).

Examples

Rectangular Matrix

Definition: A matrix where the number of rows and columns are not equal.

Examples

Zero (Null) Matrix

Definition: A matrix in which every element is zero.

Examples

[ 0  0
  0  0 ]

[ 0  0  0 ]

Identity Matrix

Definition: A square matrix with 1's on the main diagonal and 0's elsewhere (I_n).

Examples

I2 = [ 1  0
       0  1 ]

I3 = [ 1  0  0
       0  1  0
       0  0  1 ]

Diagonal Matrix

Definition: A square matrix where all non-diagonal entries are zero (only diagonal may be non-zero).

Examples

Scalar Matrix

Definition: A diagonal matrix in which every diagonal element is the same scalar value.

Examples

Triangular Matrices

Upper triangular: all entries below the main diagonal are zero. Lower triangular: all entries above the main diagonal are zero.

Upper triangular examples

Lower triangular examples

Symmetric Matrix

Definition: A square matrix equal to its transpose (A = A^T).

Examples

Skew-Symmetric Matrix

Definition: A square matrix where A^T = -A. Diagonal elements must be zero.

Examples

Singular Matrix

Definition: A square matrix with determinant equal to zero (det(A) = 0).

Examples

Non-Singular Matrix

Definition: A square matrix with non-zero determinant (det(A) ≠ 0). It has an inverse.

Examples

Orthogonal Matrix

Definition: A square matrix A such that A^TA = I (transpose is inverse).

Examples

Sparse Matrix

Definition: A matrix with mostly zero entries (only a few non-zero elements).

Examples

Quiz: Big Data Matrix

Limits of Functions of Two Variables

Definition

Intuitive Explanation

Examples

Example 1

Example 2

Example 3

Example 4

Example 5

Summary Table

Function of Two Variables – Graphical Representation

1. General 3D Graph

2. Plane Surface (z = x + y)

3. Paraboloid (z = x² + y²)

4. Saddle Surface (z = xy)

5. Contour Diagram (x² + y² = c)

Function of Three Variables

Examples of Functions of Three Variables

Graphical Representation

Level Surface Example

Cross-Section Concept

Real-Life Applications

Functions of Two and Three Variables

Section 1: Functions of Two Variables

Definition

Examples

Graphical Representation

Level Curves

Applications

Section 2: Functions of Three Variables

Definition

Examples

Graphical Representation

Applications

Section 3: Questions and Answers

Functions of Two Variables

Functions of Three Variables

Section 4: Diagrams

Unit -1

Big Data Matrix

What is a Big Data Matrix?

Characteristics of Big Data Matrix

Why Big Data Matrices are Important

Types of Data Matrix

1. Numerical Data Matrix

Definition

Example 1: Student Marks Matrix

Example 2: Weather Observation Matrix

Example 3: Sales Data Matrix

Example 4: Sensor Data Matrix

Example 5: Image Pixel Matrix

Big Data Matrix – Real World Examples

1. User–Item Matrix

2. Social Network Matrix

3. Sensor Data Matrix

4. Term–Document Matrix

5. Healthcare Data Matrix

Summary

Thank You

Integration by Partial Fractions

Type 1: Distinct linear factors

Example 1

Example 2

Type II: Repeated linear factors

Example 3

Example 4

Type III: Irreducible quadratic factors

Example 5

Row Matrix

Column Matrix

Square Matrix

Rectangular Matrix

Zero (Null) Matrix

Identity Matrix

Diagonal Matrix

Scalar Matrix

Triangular Matrices

Symmetric Matrix

Skew-Symmetric Matrix