This repository contains the notes (along with the Latex source code) taken by me at IITK during 2023-‘27.
You may clone the repository and compile any .tex file on your system as per your convenience.
Shortly, I’ll configure a workflow to compile and store PDF files after every commit. You may then download these files from the repository itself.
In the near future, you may also find some PDF files containing handwritten notes in the repository. In that case, a separate folder will be dedicated to holding these handwritten notes.
Please note that while I’ve strived for accuracy, some errors may be present.
These resources are intended for personal use and study aids; cross-reference with official course materials for verification.
Course Title: [Your Course Name Here] Instructor: [Instructor’s Name] Date: [Date]
This section covers the fundamental concepts of [Topic]. We’ll start with some key definitions and then move on to more complex ideas.
A vector field in three dimensions can be represented as a vector whose components are functions of the coordinates $(x, y, z)$. \(\mathbf{F}(x,y,z) = \langle u(x,y,z), v(x,y,z), w(x,y,z) \rangle\) The Jacobian matrix of this vector field is given by: \(J = \begin{bmatrix} \frac{\partial u}{\partial x} & \frac{\partial u}{\partial y} & \frac{\partial u}{\partial z}\\ \frac{\partial v}{\partial x} & \frac{\partial v}{\partial y} & \frac{\partial v}{\partial z}\\ \frac{\partial w}{\partial x} & \frac{\partial w}{\partial y} & \frac{\partial w}{\partial z} \end{bmatrix}\)
The Fundamental Theorem of Calculus states that if a function $f$ is continuous on the interval $[a, b]$ and $F$ is an antiderivative of $f$ on that interval, then: \(\int_a^b f(x) \,dx = F(b) - F(a)\) This is a cornerstone of calculus.
We’ll now explore more advanced topics, including the Divergence Theorem and Stokes’ Theorem.
The Divergence Theorem relates the flux of a vector field out of a closed surface to the divergence of the field in the volume enclosed by the surface. \(\iiint_V (\nabla \cdot \mathbf{F}) \,dV = \oiint_{\partial V} (\mathbf{F} \cdot \mathbf{n}) \,dS\) The term $\nabla \cdot \mathbf{F}$ is the divergence of the vector field $\mathbf{F}$.
Stokes’ Theorem relates the line integral of a vector field around a closed loop to the surface integral of the curl of the field over any surface bounded by that loop. \(\oint_{\partial S} \mathbf{F} \cdot d\mathbf{r} = \iint_S (\nabla \times \mathbf{F}) \cdot d\mathbf{S}\)
End of Notes