4.3 Vector Analysis, Linear Algebra, Matrices & Complex Analysis

4.3 Vector Analysis, Linear Algebra, Matrices & Complex Analysis

Advanced engineering mathematics unifies spatial electromagnetic field theory, multi-variable circuit network equations, system dynamics, and complex boundary integrals. For electrical engineers preparing for the PRC Board Exam, proficiency in vector calculus differential and integral operators, matrix transformations (Cramer's Rule and eigenvalue analysis), and complex analytical residue integration provides essential tools for electromagnetic fields, power system stability, and advanced circuit synthesis.

---

1. Vector Operations & Differential Calculus

Vector Multiplication

• Dot (Scalar) Product: $\mathbf{A} \cdot \mathbf{B} = |\mathbf{A}||\mathbf{B}| \cos\theta = A_x B_x + A_y B_y + A_z B_z$
• Cross (Vector) Product: $\mathbf{A} \times \mathbf{B} = |\mathbf{A}||\mathbf{B}| \sin\theta \, \hat{\mathbf{n}} = \det \begin{bmatrix} \hat{\mathbf{i}} & \hat{\mathbf{j}} & \hat{\mathbf{k}} \\ A_x & A_y & A_z \\ B_x & B_y & B_z \end{bmatrix}$
• Scalar Triple Product: $\mathbf{A} \cdot (\mathbf{B} \times \mathbf{C}) = \det \begin{bmatrix} A_x & A_y & A_z \\ B_x & B_y & B_z \\ C_x & C_y & C_z \end{bmatrix}$ (Represents parallelepiped volume)
• Vector Triple Product: $\mathbf{A} \times (\mathbf{B} \times \mathbf{C}) = (\mathbf{A} \cdot \mathbf{C}) \mathbf{B} - (\mathbf{A} \cdot \mathbf{B}) \mathbf{C}$ (BAC-CAB rule)

Vector Differential Operators (Cartesian Coordinates)

Using Del operator $\nabla = \hat{\mathbf{i}}\frac{\partial}{\partial x} + \hat{\mathbf{j}}\frac{\partial}{\partial y} + \hat{\mathbf{k}}\frac{\partial}{\partial z}$:

1. Gradient of a Scalar Field $\phi$:

   $$\nabla \phi = \frac{\partial \phi}{\partial x} \hat{\mathbf{i}} + \frac{\partial \phi}{\partial y} \hat{\mathbf{j}} + \frac{\partial \phi}{\partial z} \hat{\mathbf{k}}$$

   Physical Meaning: Points in the direction of maximum rate of increase of scalar potential $\phi$ (e.g., Electric Field $\mathbf{E} = -\nabla V$).

2. Divergence of a Vector Field $\mathbf{A}$:

   $$\nabla \cdot \mathbf{A} = \frac{\partial A_x}{\partial x} + \frac{\partial A_y}{\partial y} + \frac{\partial A_z}{\partial z}$$

   Physical Meaning: Measures net outward flux per unit volume from an infinitesimal region. If $\nabla \cdot \mathbf{A} = 0$, field $\mathbf{A}$ is solenoidal (e.g., Magnetic Field $\nabla \cdot \mathbf{B} = 0$).

3. Curl of a Vector Field $\mathbf{A}$:

   $$\nabla \times \mathbf{A} = \left( \frac{\partial A_z}{\partial y} - \frac{\partial A_y}{\partial z} \right) \hat{\mathbf{i}} + \left( \frac{\partial A_x}{\partial z} - \frac{\partial A_z}{\partial x} \right) \hat{\mathbf{j}} + \left( \frac{\partial A_y}{\partial x} - \frac{\partial A_x}{\partial y} \right) \hat{\mathbf{k}}$$

   Physical Meaning: Measures local circulation or rotational intensity per unit area. If $\nabla \times \mathbf{A} = \mathbf{0}$, field $\mathbf{A}$ is irrotational or conservative.

4. Laplacian Operator $\nabla^2$:

   $$\nabla^2 \phi = \nabla \cdot (\nabla \phi) = \frac{\partial^2 \phi}{\partial x^2} + \frac{\partial^2 \phi}{\partial y^2} + \frac{\partial^2 \phi}{\partial z^2}$$

---

2. Integral Theorems of Vector Calculus

Gauss's Divergence Theorem

Relates a volume integral of field divergence over region $V$ to a surface flux integral over bounding closed surface $S$:

$$\iiint_V (\nabla \cdot \mathbf{A}) \, dV = \iint_S \mathbf{A} \cdot d\mathbf{S}$$

Stokes' Theorem

Relates a surface integral of field curl over open surface $S$ to a closed line integral along bounding perimeter curve $C$:

$$\iint_S (\nabla \times \mathbf{A}) \cdot d\mathbf{S} = \oint_C \mathbf{A} \cdot d\mathbf{r}$$

Green's Theorem in the Plane

Special 2D case of Stokes' Theorem for planar region $R$ bounded by simple closed curve $C$:

$$\oint_C (M \, dx + N \, dy) = \iint_R \left( \frac{\partial N}{\partial x} - \frac{\partial M}{\partial y} \right) dA$$

---

3. Linear Algebra, Matrices & Cramer's Rule

Matrix Inversion

For an non-singular $n \times n$ square matrix $\mathbf{A}$ with $\det(\mathbf{A}) \neq 0$:

$$\mathbf{A}^{-1} = \frac{1}{\det(\mathbf{A})} \text{adj}(\mathbf{A})$$

where $\text{adj}(\mathbf{A}) = \mathbf{C}^T$ is the transpose of the cofactor matrix $\mathbf{C}$.

Cramer's Rule for Solving Linear Systems

For a system of $n$ linear equations $\mathbf{A} \mathbf{x} = \mathbf{b}$:

$$x_i = \frac{\det(\mathbf{A}_i)}{\det(\mathbf{A})} \quad (i = 1, 2, \dots, n)$$

where matrix $\mathbf{A}_i$ is formed by replacing the $i$-th column of coefficient matrix $\mathbf{A}$ with column vector $\mathbf{b}$.

---

4. Eigenvalues & Eigenvectors

For a square matrix $\mathbf{A}$, non-zero eigenvector $\mathbf{v}$ and scalar eigenvalue $\lambda$ satisfy:

$$\mathbf{A} \mathbf{v} = \lambda \mathbf{v} \implies (\mathbf{A} - \lambda \mathbf{I}) \mathbf{v} = \mathbf{0}$$

Characteristic Equation

To obtain non-trivial solutions $\mathbf{v} \neq \mathbf{0}$:

$$\det(\mathbf{A} - \lambda \mathbf{I}) = 0$$

Key Eigenvalue Properties

• Trace Identity: Sum of eigenvalues equals the matrix trace: $\sum_{i=1}^n \lambda_i = \text{Tr}(\mathbf{A}) = \sum_{i=1}^n a_{ii}$.
• Determinant Identity: Product of eigenvalues equals matrix determinant: $\prod_{i=1}^n \lambda_i = \det(\mathbf{A})$.
• System Stability: State space matrix $\dot{\mathbf{x}} = \mathbf{A}\mathbf{x}$ is asymptotically stable if and only if all eigenvalues $\lambda_i$ have strictly negative real parts ($\text{Re}\{\lambda_i\} < 0$).

---

5. Complex Variable Theory & Residue Calculus

Analytic Functions & Cauchy-Riemann Equations

A complex function $f(z) = u(x, y) + j v(x, y)$ is analytic (differentiable) in a region if continuous first partial derivatives satisfy the Cauchy-Riemann Equations:

$$\frac{\partial u}{\partial x} = \frac{\partial v}{\partial y} \quad \text{and} \quad \frac{\partial u}{\partial y} = -\frac{\partial v}{\partial x}$$

Both $u(x,y)$ and $v(x,y)$ satisfy Laplace's equation ($\nabla^2 u = 0, \nabla^2 v = 0$) and are called harmonic conjugates.

Cauchy's Integral Theorem & Formula

• Cauchy's Theorem: If $f(z)$ is analytic inside and on simple closed curve $C$: $$\oint_C f(z) \, dz = 0$$
• Cauchy's Integral Formula: If $f(z)$ is analytic inside $C$ containing point $z_0$: $$\oint_C \frac{f(z)}{z - z_0} \, dz = 2\pi j \cdot f(z_0)$$

Cauchy's Residue Theorem

For a function $f(z)$ with isolated singular points (poles) $z_1, z_2, \dots, z_k$ enclosed by contour $C$:

$$\oint_C f(z) \, dz = 2\pi j \sum_{i=1}^k \text{Res}(f, z_i)$$

For a simple pole at $z = z_0$, the residue is calculated as:

$$\text{Res}(f, z_0) = \lim_{z \to z_0} (z - z_0) f(z)$$

---

Step-by-Step Worked Numerical Examples

Example 4.3.1: Vector Divergence & Flux Calculation via Gauss's Theorem

Problem: Evaluate the outward flux of vector field $\mathbf{A} = 2x \hat{\mathbf{i}} + 3y \hat{\mathbf{j}} + 4z \hat{\mathbf{k}}$ through the closed surface of a sphere $x^2 + y^2 + z^2 = 9$ (radius $R = 3$).

Solution:

1. Calculate the divergence of vector field $\mathbf{A}$: $$\nabla \cdot \mathbf{A} = \frac{\partial}{\partial x}(2x) + \frac{\partial}{\partial y}(3y) + \frac{\partial}{\partial z}(4z) = 2 + 3 + 4 = 9$$
2. Apply Gauss's Divergence Theorem: $$\text{Flux} = \iint_S \mathbf{A} \cdot d\mathbf{S} = \iiint_V (\nabla \cdot \mathbf{A}) \, dV = \iiint_V 9 \, dV = 9 \iiint_V dV$$
3. Compute sphere volume $V$: $$V = \frac{4}{3} \pi R^3 = \frac{4}{3} \pi (3)^3 = \frac{4}{3} \pi (27) = 36\pi$$
4. Calculate total flux: $$\text{Flux} = 9 \cdot (36\pi) = 324\pi \approx 1017.876 \text{ units}$$

---

Example 4.3.2: Multi-Mesh Circuit Solution via Cramer's Rule

Problem: Solve for mesh currents $I_1$ and $I_2$ in the 2-mesh network system:

$$\begin{bmatrix} 5 & -2 \\ -2 & 4 \end{bmatrix} \begin{bmatrix} I_1 \\ I_2 \end{bmatrix} = \begin{bmatrix} 12 \\ 0 \end{bmatrix}$$

Solution:

1. Calculate system determinant $D = \det(\mathbf{A})$: $$D = \det \begin{bmatrix} 5 & -2 \\ -2 & 4 \end{bmatrix} = (5)(4) - (-2)(-2) = 20 - 4 = 16$$
2. Formulate modified matrix $\mathbf{A}_1$ for $I_1$: $$\mathbf{A}_1 = \begin{bmatrix} 12 & -2 \\ 0 & 4 \end{bmatrix} \implies D_1 = \det(\mathbf{A}_1) = (12)(4) - (-2)(0) = 48$$
3. Formulate modified matrix $\mathbf{A}_2$ for $I_2$: $$\mathbf{A}_2 = \begin{bmatrix} 5 & 12 \\ -2 & 0 \end{bmatrix} \implies D_2 = \det(\mathbf{A}_2) = (5)(0) - (12)(-2) = 24$$
4. Apply Cramer's Rule: $$I_1 = \frac{D_1}{D} = \frac{48}{16} = 3.0 \text{ A}$$ $$I_2 = \frac{D_2}{D} = \frac{24}{16} = 1.5 \text{ A}$$

---

Example 4.3.3: System Matrix Eigenvalue Analysis

Problem: Determine the eigenvalues $\lambda_1$ and $\lambda_2$ for state dynamic matrix $\mathbf{A} = \begin{bmatrix} -4 & 2 \\ 1 & -3 \end{bmatrix}$ and assess system stability.

Solution:

1. Formulate characteristic equation $\det(\mathbf{A} - \lambda \mathbf{I}) = 0$: $$\det \begin{bmatrix} -4 - \lambda & 2 \\ 1 & -3 - \lambda \end{bmatrix} = 0$$
2. Expand determinant quadratic: $$(-4 - \lambda)(-3 - \lambda) - (2)(1) = 0$$ $$\lambda^2 + 7 \lambda + 12 - 2 = 0 \implies \lambda^2 + 7 \lambda + 10 = 0$$
3. Factor quadratic to find roots: $$(\lambda + 2)(\lambda + 5) = 0 \implies \lambda_1 = -2, \quad \lambda_2 = -5$$
4. Verify properties:
   • Trace check: $\text{Tr}(\mathbf{A}) = -4 + (-3) = -7$. Sum $\lambda_1 + \lambda_2 = -2 + (-5) = -7$ (Matches).
   • Determinant check: $\det(\mathbf{A}) = (-4)(-3) - (2)(1) = 10$. Product $\lambda_1 \lambda_2 = (-2)(-5) = 10$ (Matches).
5. Assess Stability: Because both eigenvalues are real and strictly negative ($\lambda_1, \lambda_2 < 0$), the dynamic system is asymptotically stable.