Vector Algebra & Vector Calculus

Classical Newtonian mechanics describes physical phenomena occurring in three-dimensional Euclidean space $\mathbb{R}^3$. Physical quantities are classified based on their transformation properties under spatial rotations and coordinate transformations.

1. Scalar and Vector Quantities

A **scalar** is a physical quantity that is invariant under spatial rotations and coordinate transformations, completely characterized by a real magnitude (and units): e.g., mass $m$, time $t$, temperature $T$, and kinetic energy $K$. A **vector** $\mathbf{A}$ is an entity possessing both magnitude and direction that transforms under a coordinate rotation according to: $$A_i' = \sum_{j=1}^3 R_{ij} A_j$$ where $R_{ij}$ is an orthogonal transformation matrix satisfying $R R^T = I$. Examples include displacement $\mathbf{r}$, velocity $\mathbf{v}$, linear momentum $\mathbf{p}$, and force $\mathbf{F}$.

2. Algebraic Operations in Cartesian Basis

In a right-handed Cartesian coordinate system with orthonormal basis vectors $\{\hat{\mathbf{i}}, \hat{\mathbf{j}}, \hat{\mathbf{k}}\}$: $$\mathbf{A} = A_x \hat{\mathbf{i}} + A_y \hat{\mathbf{j}} + A_z \hat{\mathbf{k}}, \quad \mathbf{B} = B_x \hat{\mathbf{i}} + B_y \hat{\mathbf{j}} + B_z \hat{\mathbf{k}}$$ Vector addition and scalar multiplication satisfy linear vector space axioms: $$\mathbf{A} + \mathbf{B} = (A_x + B_x)\hat{\mathbf{i}} + (A_y + B_y)\hat{\mathbf{j}} + (A_z + B_z)\hat{\mathbf{k}}$$ $$\lambda \mathbf{A} = (\lambda A_x)\hat{\mathbf{i}} + (\lambda A_y)\hat{\mathbf{j}} + (\lambda A_z)\hat{\mathbf{k}}, \quad \lambda \in \mathbb{R}$$ The magnitude (Euclidean norm) is: $$|\mathbf{A}| = A = \sqrt{A_x^2 + A_y^2 + A_z^2}$$

3. The Scalar (Dot) Product

The scalar product of two vectors is defined geometrically and algebraically as: $$\mathbf{A} \cdot \mathbf{B} = |\mathbf{A}||\mathbf{B}| \cos \theta = A_x B_x + A_y B_y + A_z B_z$$ where $\theta \in [0, \pi]$ is the angle between them.

• Orthogonality Condition: $\mathbf{A} \perp \mathbf{B} \iff \mathbf{A} \cdot \mathbf{B} = 0$.
• Projection: The scalar component of $\mathbf{A}$ along $\mathbf{B}$ is $A_B = \mathbf{A} \cdot \hat{\mathbf{B}} = \frac{\mathbf{A} \cdot \mathbf{B}}{|\mathbf{B}|}$.
• Physical Application: Mechanical work done by force $\mathbf{F}$ along displacement $d\mathbf{r}$: $dW = \mathbf{F} \cdot d\mathbf{r}$.

4. The Vector (Cross) Product

The vector product produces an axial vector perpendicular to both operands: $$\mathbf{A} \times \mathbf{B} = |\mathbf{A}||\mathbf{B}| \sin \theta \,\hat{\mathbf{n}}$$ where $\hat{\mathbf{n}}$ is determined by the right-hand rule. In determinant form: $$\mathbf{A} \times \mathbf{B} = \begin{vmatrix} \hat{\mathbf{i}} & \hat{\mathbf{j}} & \hat{\mathbf{k}} \\ A_x & A_y & A_z \\ B_x & B_y & B_z \end{vmatrix} = (A_y B_z - A_z B_y)\hat{\mathbf{i}} + (A_z B_x - A_x B_z)\hat{\mathbf{j}} + (A_x B_y - A_y B_x)\hat{\mathbf{k}}$$

• Anticommutativity: $\mathbf{B} \times \mathbf{A} = -(\mathbf{A} \times \mathbf{B})$.
• Collinearity: $\mathbf{A} \parallel \mathbf{B} \iff \mathbf{A} \times \mathbf{B} = 0$.
• Physical Application: Torque $\boldsymbol{\tau} = \mathbf{r} \times \mathbf{F}$ and angular momentum $\mathbf{L} = \mathbf{r} \times \mathbf{p}$.

Spatial fields represent physical quantities that vary continuously over space. Vector calculus describes the spatial rates of change of scalar fields $\Phi(\mathbf{r})$ and vector fields $\mathbf{V}(\mathbf{r})$.

1. The Del (Nabla) Operator

The vector differential operator $\boldsymbol{\nabla}$ in Cartesian coordinates is: $$\boldsymbol{\nabla} = \hat{\mathbf{i}} \frac{\partial}{\partial x} + \hat{\mathbf{j}} \frac{\partial}{\partial y} + \hat{\mathbf{k}} \frac{\partial}{\partial z}$$

2. Gradient of a Scalar Field ($\boldsymbol{\nabla}\Phi$)

The gradient of a differentiable scalar function $\Phi(x, y, z)$ is a vector pointing in the direction of maximum spatial increase of $\Phi$, whose magnitude equals the directional derivative: $$\boldsymbol{\nabla}\Phi = \frac{\partial \Phi}{\partial x}\hat{\mathbf{i}} + \frac{\partial \Phi}{\partial y}\hat{\mathbf{j}} + \frac{\partial \Phi}{\partial z}\hat{\mathbf{k}}$$ Physical Application: Conservative forces in mechanics are the negative gradient of their potential energy functions $U(\mathbf{r})$: $$\mathbf{F}(\mathbf{r}) = -\boldsymbol{\nabla}U(\mathbf{r})$$

3. Divergence of a Vector Field ($\boldsymbol{\nabla} \cdot \mathbf{V}$)

The divergence measures the net outflow flux per unit volume from an infinitesimal neighborhood around a point: $$\boldsymbol{\nabla} \cdot \mathbf{V} = \frac{\partial V_x}{\partial x} + \frac{\partial V_y}{\partial y} + \frac{\partial V_z}{\partial z}$$

• $\boldsymbol{\nabla} \cdot \mathbf{V} > 0$: The point is a source (net outflow).
• $\boldsymbol{\nabla} \cdot \mathbf{V} < 0$: The point is a sink (net inflow).
• $\boldsymbol{\nabla} \cdot \mathbf{V} = 0$: Solenoidal (incompressible) field.

4. Curl of a Vector Field ($\boldsymbol{\nabla} \times \mathbf{V}$)

The curl represents the microscopic circulation (vorticity) density of the vector field: $$\boldsymbol{\nabla} \times \mathbf{V} = \begin{vmatrix} \hat{\mathbf{i}} & \hat{\mathbf{j}} & \hat{\mathbf{k}} \\ \frac{\partial}{\partial x} & \frac{\partial}{\partial y} & \frac{\partial}{\partial z} \\ V_x & V_y & V_z \end{vmatrix} = \left( \frac{\partial V_z}{\partial y} - \frac{\partial V_y}{\partial z} \right)\hat{\mathbf{i}} + \left( \frac{\partial V_x}{\partial z} - \frac{\partial V_z}{\partial x} \right)\hat{\mathbf{j}} + \left( \frac{\partial V_y}{\partial x} - \frac{\partial V_x}{\partial y} \right)\hat{\mathbf{k}}$$ Crucial Theorem for Mechanics: A force field $\mathbf{F}(\mathbf{r})$ is conservative if and only if its curl vanishes everywhere in a simply-connected domain: $$\boldsymbol{\nabla} \times \mathbf{F} = 0 \iff \mathbf{F} = -\boldsymbol{\nabla}U$$