Partial Differentiation : An Introduction.

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Introduction to Partial Differentiation

In many practical situations, a quantity depends on more than one independent variable. For example, the volume of a gas may depend on both pressure and temperature, or the area of a surface may depend on two spatial coordinates. Such functions are called functions of several variables.

If a function depends on two independent variables $x$ and $y$, it is written as $$ z = f(x, y) $$

Partial differentiation is the process of finding the rate of change of a function with respect to one variable while keeping the other variable(s) constant.

The partial derivative of $z$ with respect to $x$ is denoted by $$ \frac{\partial z}{\partial x} $$ and is obtained by differentiating $z$ with respect to $x$, treating $y$ as a constant.

Similarly, the partial derivative of $z$ with respect to $y$ is denoted by $$ \frac{\partial z}{\partial y} $$ and is obtained by differentiating $z$ with respect to $y$, treating $x$ as a constant.

For example, if $$ z = x^2y + xy^2 $$ then $$ \frac{\partial z}{\partial x} = 2xy + y^2, \quad \frac{\partial z}{\partial y} = x^2 + 2xy $$

Partial differentiation plays an important role in mathematics, physics, engineering, economics, and other applied sciences. It is widely used in topics such as total derivatives, maxima and minima, Euler’s theorem, and differential equations.

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Partial Differentiation Problems with Solutions

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Problem 1

If $z = x^2y + xy^3$, find $\frac{\partial z}{\partial x}$ and $\frac{\partial z}{\partial y}$.

Solution:

$$ \frac{\partial z}{\partial x} = 2xy + y^3 $$ $$ \frac{\partial z}{\partial y} = x^2 + 3xy^2 $$

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Problem 2

If $z = x^3 + 3xy^2$, find $\frac{\partial^2 z}{\partial x^2}$ and $\frac{\partial^2 z}{\partial y^2}$.

Solution:

$$ \frac{\partial z}{\partial x} = 3x^2 + 3y^2 \Rightarrow \frac{\partial^2 z}{\partial x^2} = 6x $$ $$ \frac{\partial z}{\partial y} = 6xy \Rightarrow \frac{\partial^2 z}{\partial y^2} = 6x $$

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Problem 3

For $z = x^2y^3$, verify that $$ \frac{\partial^2 z}{\partial x \partial y} = \frac{\partial^2 z}{\partial y \partial x} $$

Solution:

$$ \frac{\partial z}{\partial x} = 2xy^3 \Rightarrow \frac{\partial^2 z}{\partial y \partial x} = 6xy^2 $$ $$ \frac{\partial z}{\partial y} = 3x^2y^2 \Rightarrow \frac{\partial^2 z}{\partial x \partial y} = 6xy^2 $$

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Problem 4

If $x^2 + y^2 + z^2 = a^2$, find $\frac{\partial z}{\partial x}$.

Solution:

$$ 2x + 2z\frac{\partial z}{\partial x} = 0 $$ $$ \frac{\partial z}{\partial x} = -\frac{x}{z} $$

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Problem 5

If $z = x^2 + y^2$, where $x = r\cos\theta$ and $y = r\sin\theta$, find $\frac{\partial z}{\partial r}$.

Solution:

$$ z = r^2(\cos^2\theta + \sin^2\theta) = r^2 $$ $$ \frac{\partial z}{\partial r} = 2r $$

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Problem 6 (Euler’s Theorem)

Verify Euler’s theorem for $z = x^2y + xy^2$.

Solution:

$$ \frac{\partial z}{\partial x} = 2xy + y^2 $$ $$ \frac{\partial z}{\partial y} = x^2 + 2xy $$ $$ x\frac{\partial z}{\partial x} + y\frac{\partial z}{\partial y} = 3x^2y + 3xy^2 = 3z $$

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Problem 7

If $z = \ln(x^2 + y^2)$, find $\frac{\partial z}{\partial x}$.

Solution:

$$ \frac{\partial z}{\partial x} = \frac{2x}{x^2 + y^2} $$

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Problem 8

If $z = e^{xy}\sin y$, find $\frac{\partial z}{\partial y}$.

Solution:

$$ \frac{\partial z}{\partial y} = e^{xy}(x\sin y + \cos y) $$

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Problem 9

If $z = x^2y + y^2$, find the total differential $dz$.

Solution:

$$ \frac{\partial z}{\partial x} = 2xy,\quad \frac{\partial z}{\partial y} = x^2 + 2y $$ $$ dz = 2xy\,dx + (x^2 + 2y)\,dy $$

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Problem 10

Find $\frac{\partial}{\partial x}(x^2y^3)$.

Solution:

$$ \frac{\partial}{\partial x}(x^2y^3) = 2xy^3 $$

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📘 End of Exam-Oriented Problems

Approximate Matrix Multiplication: Random Sampling of Columns/Rows (Sketching

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Background: Approximate Matrix Multiplication (AMM)

Matrix multiplication is a fundamental operation in scientific computing, machine learning, data mining, and numerical linear algebra. Given two matrices \( A \in \mathbb{R}^{n \times d} \) and \( B \in \mathbb{R}^{d \times m} \), the exact computation of \( AB \) requires \( O(ndm) \) arithmetic operations, which becomes computationally expensive when the inner dimension \( d \) is large.

Approximate Matrix Multiplication (AMM) addresses this challenge by computing an approximation \( \widetilde{C} \approx AB \) that is significantly faster to obtain while maintaining provable accuracy guarantees. The key idea is to reduce the dimensionality of the problem using randomized techniques, thereby lowering computational cost and memory usage.

Random Sampling of Columns and Rows (Sketching)

One of the simplest and most intuitive approaches to AMM is random sampling of columns of \( A \) and the corresponding rows of \( B \). Since matrix multiplication can be written as a sum of outer products,

\[ AB = \sum_{i=1}^{d} A_{(:,i)} B_{(i,:)} \]

we can approximate this sum by randomly selecting only a small subset of indices \( S \subset \{1,2,\dots,d\} \) with \( |S| = k \ll d \). The approximation is then given by:

\[ \widetilde{C} = \frac{d}{k} \sum_{i \in S} A_{(:,i)} B_{(i,:)} \]

The scaling factor \( \frac{d}{k} \) ensures that the estimator is unbiased, meaning:

\[ \mathbb{E}[\widetilde{C}] = AB \]

Why Sketching Works

Each column--row pair \( A_{(:,i)} B_{(i,:)} \) contributes differently to the final product. Random sampling exploits the fact that many contributions are often small or redundant, especially in large, structured, or noisy data. By sampling only a few representative pairs, we can closely approximate the full sum with much less computation.

Accuracy and Trade-offs

The accuracy of the approximation depends on the number of samples \( k \). As \( k \) increases, the approximation converges to the exact product. With appropriate sampling strategies (such as importance sampling based on column norms), strong error bounds can be obtained in Frobenius or spectral norm.

Random sampling of columns and rows is widely used as a building block in:

• Large-scale machine learning algorithms
• Dimensionality reduction and feature compression
• Recommender systems
• Streaming and distributed computation

In the following examples, we illustrate how this technique works through simple numerical demonstrations.

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Approximate Matrix Multiplication using Random Sampling of Columns/Rows

Given matrices \( A \in \mathbb{R}^{n \times d} \) and \( B \in \mathbb{R}^{d \times m} \), approximate:

\[ AB \approx \frac{d}{k} \sum_{i \in S} A_{(:,i)} B_{(i,:)} \]

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Example 1: Vector × Vector

\( A = [1 \; 2 \; 3 \; 4] \), \( B = [5 \; 6 \; 7 \; 8]^T \)

Exact:

\[ AB = 70 \]

Sample: \( S = \{2,4\} \), \( d=4, k=2 \)

Approximate:

\[ AB \approx 2(2\cdot6 + 4\cdot8) = 88 \]

Example 2: Matrix × Vector

\[ A = \begin{bmatrix} 1 & 3 & 5\\ 2 & 4 & 6 \end{bmatrix}, \quad B = \begin{bmatrix} 1\\2\\3 \end{bmatrix} \]

Exact:

\[ AB = \begin{bmatrix} 22\\28 \end{bmatrix} \]

Sample: \( S=\{3\} \), \( d=3 \)

Approximate:

\[ AB \approx 3 \begin{bmatrix} 5\\6 \end{bmatrix} \cdot 3 = \begin{bmatrix} 45\\54 \end{bmatrix} \]

Example 3: Matrix × Matrix

\[ A = \begin{bmatrix} 1 & 2 & 3\\ 4 & 5 & 6 \end{bmatrix}, \quad B = \begin{bmatrix} 7 & 8\\ 9 & 10\\ 11 & 12 \end{bmatrix} \]

Exact:

\[ AB = \begin{bmatrix} 58 & 64\\ 139 & 154 \end{bmatrix} \]

Sample: \( S=\{1,3\} \)

Approximate:

\[ AB \approx 1.5(A_{(:,1)}B_{(1,:)} + A_{(:,3)}B_{(3,:)}) \]

Example 4: Single Column Sampling

\[ A = [2 \; 4 \; 6], \quad B = [1 \; 3 \; 5]^T \]

Exact: \( 44 \)

Approximate:

\[ AB \approx 3(4\cdot3) = 36 \]

Example 5: Symmetric Vectors

\[ A = B = [1 \; 2 \; 3] \]

Exact: \( 14 \)

Approximate:

\[ AB \approx \frac{3}{2}(1^2 + 3^2) = 15 \]

Example 6: Tall Matrix × Vector

\[ A = \begin{bmatrix} 1 & 0 & 2\\ 0 & 1 & 3\\ 1 & 1 & 1 \end{bmatrix}, \quad B = \begin{bmatrix} 2\\4\\6 \end{bmatrix} \]

Approximate:

\[ AB \approx 3 \begin{bmatrix} 2\\3\\1 \end{bmatrix} \cdot 6 \]

Example 7: Larger Matrix × Matrix

\[ A = \begin{bmatrix} 1 & 0 & 2 & 1\\ 2 & 1 & 0 & 3 \end{bmatrix}, \quad B = \begin{bmatrix} 1 & 2\\ 3 & 4\\ 5 & 6\\ 7 & 8 \end{bmatrix} \]

Approximate:

\[ AB \approx 2(A_{(:,1)}B_{(1,:)} + A_{(:,4)}B_{(4,:)}) \]

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Error Bounds and Intuition

The approximation quality in Random Sampling–based Approximate Matrix Multiplication depends on how many column–row pairs are sampled. Let \( \widetilde{C} \) denote the approximate product and \( C = AB \) the exact product.

With uniform random sampling, the expected error measured in Frobenius norm satisfies:

\[ \mathbb{E}\left[\| \widetilde{C} - C \|_F^2 \right] \le \frac{d}{k} \sum_{i=1}^{d} \|A_{(:,i)}\|_2^2 \|B_{(i,:)}\|_2^2 \]

This inequality shows that the error decreases as the number of samples \( k \) increases. In particular, doubling the number of sampled columns roughly halves the expected error.

Intuition Behind the Error Bound

Matrix multiplication can be viewed as summing many outer products. Random sampling replaces this full sum with an estimate based on only a few terms. The approximation error arises from ignoring some column–row pairs. If most of these pairs contribute little to the final product, then the error remains small.

Thus, matrices with redundant, correlated, or low-energy columns are especially well-suited for approximate multiplication.

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Importance Sampling

Uniform sampling treats all column–row pairs equally. However, in practice, some columns of \( A \) and rows of \( B \) contribute much more to \( AB \) than others. Importance sampling improves accuracy by sampling indices with higher probability when they have larger impact.

A common choice of sampling probability is:

\[ p_i = \frac{\|A_{(:,i)}\|_2 \|B_{(i,:)}\|_2} {\sum_{j=1}^{d} \|A_{(:,j)}\|_2 \|B_{(j,:)}\|_2} \]

The approximation becomes:

\[ \widetilde{C} = \frac{1}{k} \sum_{i \in S} \frac{1}{p_i} A_{(:,i)} B_{(i,:)} \]

Importance sampling significantly reduces variance and provides tighter error bounds, especially when column norms vary widely.

Key idea: sample what matters most.

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Real-World Machine Learning Applications

• Recommendation Systems: User–item matrices are often large and sparse. Approximate matrix multiplication accelerates similarity computations and matrix factorization.
• Deep Learning: Large fully connected layers and attention mechanisms can be approximated using sketching techniques to reduce training and inference cost.
• Natural Language Processing: Word embeddings and document–term matrices benefit from approximate multiplication during similarity search and latent semantic analysis.
• Kernel Methods: AMM is used to approximate large kernel matrices in support vector machines and Gaussian processes.
• Large-Scale Linear Regression: Sketching reduces the cost of computing \( X^T X \) and \( X^T y \) in high-dimensional data.

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Conclusion: Random sampling of columns and rows significantly reduces computation while providing an unbiased approximation. Increasing the sample size improves accuracy.

Test - 1

Quiz: Big Data Matrix

Instructions: Enter your details, answer all questions, and click Submit Quiz.

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Name:

Roll Number:

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1. What is a Big Data Matrix?

A. A small table stored in Excel
B. A large-scale matrix representation of big data
C. A graphical visualization tool
D. A database index

2. In an Object–Attribute Matrix, rows represent ______.

A. Attributes
B. Columns
C. Objects or entities
D. Algorithms

3. Which matrix is commonly used in recommendation systems?

A. Document–Term Matrix
B. Object–Attribute Matrix
C. User–Item Matrix
D. Time-Series Matrix

4. A Sparse Matrix is characterized by:

A. All values are non-zero
B. Mostly negative values
C. Mostly zero values
D. Equal rows and columns

5. Document–Term Matrix is mainly used in:

A. Image processing
B. Network routing
C. Natural Language Processing
D. Sorting algorithms

6. Which matrix represents relationships between nodes?

A. Time-Series Matrix
B. Sparse Matrix
C. Adjacency Matrix
D. Object–Attribute Matrix

7. Why are Big Data Matrices often sparse?

A. Data is random
B. Most entities have few interactions
C. Storage systems force zeros
D. All features are correlated

8. Which technology is used to process Big Data Matrices?

A. MS Word
B. Apache Spark
C. Notepad
D. PowerPoint

9. In a Time-Series Matrix, rows represent:

A. Features
B. Objects
C. Time points
D. Classes

10. The most basic form of data matrix is:

A. Sparse Matrix
B. Graph Matrix
C. Object–Attribute Matrix
D. Document–Term Matrix

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Prepared by: Brajesh Jha

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)

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1. General 3D Graph

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

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

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2. Plane Surface (z = x + y)

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

Application: Cost, temperature variation.

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3. Paraboloid (z = x² + y²)

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

Application: Heat distribution, potential energy.

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4. Saddle Surface (z = xy)

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

Application: Profit–loss analysis.

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5. Contour Diagram (x² + y² = c)

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

Application: Topographic maps, weather maps.

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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)

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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

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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.

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Cross-Section Concept

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

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

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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)

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

Assignment

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

 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

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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

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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

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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

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Types of Data Matrix

1. Numerical Data Matrix
2. Binary Data Matrix
3. Categorical Data Matrix
4. Sparse Data Matrix
5. Distributed Data Matrix

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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
• $x_{ij} \in \mathbb{R}$

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Example 1: Student Marks Matrix

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

Average Marks of Student i:

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

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Example 2: Weather Observation Matrix

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

Mean Temperature:

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

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Example 3: Sales Data Matrix

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

Total Sales:

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

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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:

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

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Example 5: Image Pixel Matrix

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

Normalization:

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

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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

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Summary

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

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Thank You