1 Vector Calculus
1.1 Overview
- Preliminaries:
- Linear Algebra
- Geometry of \(\mathbb{R}^n\):
- Dot product
- Norm and distance
- Orthogonality
- Projections
- Level sets and surfaces
- Calculus
- Linear Algebra
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- Hessian
- Differential Operators:
- Divergence
- Curl
- Laplacian
- Line and Surface Integrals
- Fundamental Theorems:
- Fundamental Theorem for Line Integrals
- Green Theorem
- Stoke Theorem
- Divergence Theorem
- Coordinate Systems
- Polar coordinates
- Cylindrical coordinates
- Spherical coordinates
- Jacobians of transformations
- Vector Calculus in Physics
- Maxwell’s equations
- Fluid dynamics (divergence = incompressibility)
- Potential theory
- Heat and wave equations
1.2 Vector Fields in Euclidean Space
A vector field assigns a vector to each point in space. In \(\mathbb{R}^n\), a vector field is a function
\[ \mathbf{F} : \mathbb{R}^n \to \mathbb{R}^m \]
commonly with \(m = n\), so
\[ \mathbf{F}(x) = \begin{pmatrix} F_1(x) \\ \vdots \\ F_n(x) \end{pmatrix}. \]
Example: In three dimensions,
\[ \mathbf{F}(x,y,z) = (F_1(x,y,z), F_2(x,y,z), F_3(x,y,z)). \]
Intuitively, a vector field can be thought of as assigning a vector (objects with both size and direction) to every point in space. It describes the change of vectors as it moves through the space, or the flow. Some common examples from physics include:
- Velocity of a fluid.
- Force acting at each point.
- Wind direction and speed.
- Wlectric or magnetic fields.
Each location has both direction and magnitude.
Example: A radial outward field in \(\mathbb{R}^2\) is described by
\[ \mathbf{F}(x,y) = (x,y). \]
Vectors point away from the origin and grow in magnitude with distance.
1.3 Gradients of Vector Fields
The gradient of a scalar-valued differentiable function \(f\) of several variables is the vector field \(\nabla f\) whose value at a point \(p\) gives the direction and the rate of fastest increase.
In \(n\)-dimensional Cartesian coordinates with Euclidean metric the gradient, if it exists, is given by \[ \nabla f:=\sum_{i=1}^{n} \frac{{\partial f}}{\partial x_{i}}\boldsymbol{i}_{i}, \] where \(\boldsymbol{i}_{j}\) is the standard unit vector in the directions of the \(x_{j}\) coordinate respectively.
This result is written in standard basis notation but alternatively could be written in column vector notation as \[ \nabla f:=\begin{bmatrix} \frac{{\partial f}}{\partial x_{1}} \\ \vdots \\ \frac{{\partial f}}{\partial x_{n}} \end{bmatrix}=\left[ \frac{{\partial f}}{\partial x_{i}} \right]_{i=1,..,n}. \]
Example (Cartesian Gradient): We consider the function \(f(x,y,z)=2x+3xy^2-\sin (z)\). The gradient is \[ \nabla f:=\begin{bmatrix} 2+3y^2 \\ 6xy \\ -\cos(z) \end{bmatrix}. \]
Considering general coordinates in an \(n\)-dimensional space \(x=(x^i)_{i=1,\dots,n}\) and let \(f\) be a scalar-valued differentiable function \(f(x):=f(x^1, \dots, x^n)\). The gradient of \(f\) is given by
\[ \nabla f= \frac{{\partial f}}{\partial x^i}g^{ij}\boldsymbol{e}_{j}, \] where \(\boldsymbol{e}^i=dx^i\) and \(\boldsymbol{e}_{i}=\frac{{\partial \boldsymbol{x}}}{\partial x^i}\) refer to the unnormalized local covariant and contravariant bases respectively, \(g^{i,j}\) is the inverse metric tensor, and the Einstein summation convention implies summation over \(i\) and \(j\).
1.3.1 Gradient Results
1.3.1.1 Relationship to Total Derivative
The gradient is closely related to the total derivative \(df\) in that they are transpose of one another \[ \nabla f=df^T \]
1.3.1.2 Conservative Vector Fields
A conservative vector field is a vector field that is the gradient of a scalar potential function \(f\)
\[ F = \nabla f \]
Key properties include path-independent line integrals, zero circulation around closed loops
\[ \oint F\cdot dr = 0 \]
Furthermore, for 3D fields, conservative vector fields have a vanishing curl
\[ \nabla\times F = 0 \]
1.3.1.3 Directional Derivative
\[ D_v f = \nabla f \cdot v. \]
1.3.1.4 Steepest Ascent
\[ \max_{\|v\|=1} D_v f = \|\nabla f\|. \]
1.4 Jacobian
The Jacobian matrix is the natural generalization of the derivative to vector functions.
For \(\mathbf{F} : \mathbb{R}^n \to \mathbb{R}^m\) the Jacobian is $$ J_{}=()_{i,j}
. $$
Clearly we have the following relationship between the Jacobian and the gradient
\[ J_F = (\nabla F)^{\top}. \]
1.4.1 Jacobian Results
1.4.1.1 Linearization
Linearization is the process of approximatin
\[ \mathbf{F}(x+h) = \mathbf{F}(x) + J_{\mathbf{F}}(x) h + o(\|h\|). \]
1.5 Change of Variables
If \(T\) is invertible:
\[ \int_{T(U)} f(y)\, dy = \int_U f(T(x)) \left| \det J_T(x) \right| dx. \]
2 5. Hessian
For scalar field \(f\):
\[ H_f = \left( \frac{\partial^2 f}{\partial x_i \partial x_j} \right). \]
2.1 Second-Order Taylor Expansion
\[ f(x+h) = f(x) + \nabla f(x)\cdot h + \frac{1}{2} h^T H_f(x) h + o(\|h\|^2). \]
2.2 Classification of Critical Points
If
\[ \nabla f(x_0) = 0, \]
then:
- \(H_f\) positive definite → local minimum
- \(H_f\) negative definite → local maximum
- \(H_f\) indefinite → saddle point
3 6. Differential Operators
3.1 Divergence
\[ \nabla \cdot \mathbf{F} = \sum_{i=1}^n \frac{\partial F_i}{\partial x_i}. \]
3.2 Curl (in \(\mathbb{R}^3\))
\[ \nabla \times \mathbf{F} = \begin{vmatrix} \mathbf{i} & \mathbf{j} & \mathbf{k} \\ \partial_x & \partial_y & \partial_z \\ F_1 & F_2 & F_3 \end{vmatrix}. \]
3.3 Laplacian
\[ \Delta f = \nabla \cdot \nabla f = \sum_{i=1}^n \frac{\partial^2 f}{\partial x_i^2}. \]
Heat equation:
\[ u_t = \Delta u. \]
Laplace equation:
\[ \Delta u = 0. \]
4 7. Line and Surface Integrals
4.1 Line Integral
If \(\gamma : [a,b] \to \mathbb{R}^n\) parametrizes \(C\):
\[ \int_C \mathbf{F} \cdot d\mathbf{r} = \int_a^b \mathbf{F}(\gamma(t)) \cdot \gamma'(t) \, dt. \]
4.2 Surface Integral (Flux)
\[ \iint_S \mathbf{F} \cdot \mathbf{n} \, dS. \]
If \(\mathbf{r}(u,v)\) parametrizes \(S\):
\[ \mathbf{n}\, dS = \partial_u \mathbf{r} \times \partial_v \mathbf{r} \, du dv. \]
5 8. Fundamental Theorems
5.1 Fundamental Theorem for Line Integrals
If \(\mathbf{F} = \nabla f\):
\[ \int_C \mathbf{F} \cdot d\mathbf{r} = f(\text{end}) - f(\text{start}). \]
5.2 Green’s Theorem
\[ \oint_C P\, dx + Q\, dy = \iint_D \left( \frac{\partial Q}{\partial x} - \frac{\partial P}{\partial y} \right) dA. \]
5.3 Stokes’ Theorem
\[ \oint_{\partial S} \mathbf{F} \cdot d\mathbf{r} = \iint_S (\nabla \times \mathbf{F}) \cdot \mathbf{n} \, dS. \]
5.4 Divergence Theorem
\[ \iiint_V \nabla \cdot \mathbf{F} \, dV = \iint_{\partial V} \mathbf{F} \cdot \mathbf{n} \, dS. \]
6 9. Coordinate Systems
6.1 Polar Coordinates
\[ x = r \cos \theta \]
\[ y = r \sin \theta \]
\[ dA = r\, dr\, d\theta. \]
6.2 Cylindrical Coordinates
\[ x = r\cos\theta, \quad y = r\sin\theta, \quad z = z. \]
\[ dV = r\, dr\, d\theta\, dz. \]
6.3 Spherical Coordinates
\[ x = \rho \sin\phi \cos\theta \]
\[ y = \rho \sin\phi \sin\theta \]
\[ z = \rho \cos\phi \]
\[ dV = \rho^2 \sin\phi \, d\rho \, d\phi \, d\theta. \]
7 10. Vector Calculus in Physics
7.1 Maxwell’s Equations
\[ \nabla \cdot \mathbf{E} = \frac{\rho}{\varepsilon_0} \]
\[ \nabla \cdot \mathbf{B} = 0 \]
\[ \nabla \times \mathbf{E} = -\frac{\partial \mathbf{B}}{\partial t} \]
\[ \nabla \times \mathbf{B} = \mu_0 \mathbf{J} + \mu_0 \varepsilon_0 \frac{\partial \mathbf{E}}{\partial t}. \]
7.2 Fluid Dynamics (Incompressibility)
\[ \nabla \cdot \mathbf{v} = 0. \]
7.3 Potential Theory
\[ \Delta u = 0. \]
7.4 Heat Equation
\[ u_t = \Delta u. \]
7.5 Wave Equation
\[ u_{tt} = c^2 \Delta u. \]