1: Vector calculus

Section 1: Foundations – Vector Calculus

Vector calculus extends basic calculus to vector fields, providing essential tools for describing physical phenomena like fluid flow and electromagnetism. Mastery of vector operations and their geometric interpretations is crucial.

1. Vector Operations:

• Dot Product (Scalar Product): $\vec{A} \cdot \vec{B} = |\vec{A}||\vec{B}|\cos\theta = A_xB_x + A_yB_y + A_zB_z$. It measures projection and work done. Orthogonality: $\vec{A} \cdot \vec{B} = 0$.
• Cross Product (Vector Product): $\vec{A} \times \vec{B} = \begin{vmatrix} \hat{i} & \hat{j} & \hat{k} \\ A_x & A_y & A_z \\ B_x & B_y & B_z \end{vmatrix} = (A_yB_z - A_zB_y)\hat{i} + (A_zB_x - A_xB_z)\hat{j} + (A_xB_y - A_yB_x)\hat{k}$. Result is perpendicular to $\vec{A}$ and $\vec{B}$ (right-hand rule), magnitude $|\vec{A}||\vec{B}|\sin\theta$ equals area of parallelogram spanned. Parallelism: $\vec{A} \times \vec{B} = \vec{0}$.
• Scalar Triple Product: $\vec{A} \cdot (\vec{B} \times \vec{C}) = \vec{B} \cdot (\vec{C} \times \vec{A}) = \vec{C} \cdot (\vec{A} \times \vec{B})$. Represents volume of parallelepiped. Vectors coplanar if zero.

2. Differential Operators:

• Gradient ($\nabla f$): For scalar field $f(x,y,z)$, $\nabla f = \frac{\partial f}{\partial x}\hat{i} + \frac{\partial f}{\partial y}\hat{j} + \frac{\partial f}{\partial z}\hat{k}$. Points toward steepest ascent; a vector field.
• Divergence ($\nabla \cdot \vec{F}$): For vector field $\vec{F} = F_x\hat{i} + F_y\hat{j} + F_z\hat{k}$, $\nabla \cdot \vec{F} = \frac{\partial F_x}{\partial x} + \frac{\partial F_y}{\partial y} + \frac{\partial F_z}{\partial z}$. Measures "source strength" (flux per unit volume) at a point. Scalar result.
• Curl ($\nabla \times \vec{F}$): $\nabla \times \vec{F} = \begin{vmatrix} \hat{i} & \hat{j} & \hat{k} \\ \partial/\partial x & \partial/\partial y & \partial/\partial z \\ F_x & F_y & F_z \end{vmatrix}$. Measures local rotation/circulation density ("curliness"). Vector result. Irrotational field if $\nabla \times \vec{F} = \vec{0}$ (implies $\vec{F} = \nabla \phi$).

3. Integral Theorems (Relate derivatives to integrals):

• Gauss's Divergence Theorem: Relates flux through a closed surface $S$ to divergence within enclosed volume $V$: $\oiint_S \vec{F} \cdot d\vec{S} = \iiint_V (\nabla \cdot \vec{F}) dV$ Essential for electromagnetism (Gauss's Law) and fluid dynamics.
• Stokes' Theorem: Relates circulation around a closed curve $C$ to curl through any surface $S$ bounded by $C$: $\oint_C \vec{F} \cdot d\vec{r} = \iint_S (\nabla \times \vec{F}) \cdot d\vec{S}$ Key for Faraday's Law and conservative fields ($\oint \vec{F} \cdot d\vec{r} = 0$ if $\nabla \times \vec{F} = \vec{0}$).

4. Line and Surface Integrals:

• Line Integral (Work): $\int_C \vec{F} \cdot d\vec{r} = \int_a^b \vec{F}(\vec{r}(t)) \cdot \vec{r}'(t) dt$. Computes work done by $\vec{F}$ along path $C$.
• Surface Integral (Flux): $\iint_S \vec{F} \cdot d\vec{S} = \iint_D \vec{F}(\vec{r}(u,v)) \cdot (\vec{r}_u \times \vec{r}_v) du dv$. Computes flux of $\vec{F}$ through surface $S$.