Dr. Brajesh Kumar Jha

Geometric Interpretation of Total Derivative Consider a function of two variables $$ z = f(x, y) $$ This function represents a surface in three-dimensional space. Each point on the surface corresponds to a particular value of $x$ and $y$. x z y (x, y, z) The red point represents a point on the surface corresponding to the values $(x, y)$. Small Changes in Variables Suppose the independent variables $x$ and $y$ change by small amounts $dx$ and $dy$. These small changes cause a change in $z$, denoted by $dz$. dx dy The movement along the $x$-direction by $dx$ and along the $y$-direction by $dy$ together produce a total change in the value of $z$. ============================= Total Change in z The total change in $z$ is the combined effect of the change due to $x$ and the change due to $y$. Mathematically, it is given by: $$ dz = \frac{\partial z}{\partial x}\,dx + \frac{\partial z}{\partial ...

============================================================================================= 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 $$ \frac{\partial z}{\partial y} $$ a...

Approximate Matrix Multiplication using Random Sampling 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 ra...

Quiz: Big Data Matrix Instructions: Enter your details, answer all questions, and click Submit Quiz . Name: Roll Number: 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...

Limits of Functions 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) ...

  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....

 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 .        ...