Physics: Vectors and Vector operations

Physics

Vectors

• Vectors can be added analytically by summing their respective components.
• If vector $\overrightarrow{A} = A_x\hat{i} + A_y\hat{j} + A_z\hat{k}$ and vector $\overrightarrow{B} = B_x\hat{i} + B_y\hat{j} + B_z\hat{k}$, then $\overrightarrow{A} + \overrightarrow{B} = (A_x + B_x)\hat{i} + (A_y + B_y)\hat{j} + (A_z + B_z)\hat{k}$.

Scalar Product (Dot Product)

• The dot product of two vectors is equal to the product of their magnitudes and the cosine of the angle between them: $\overrightarrow{A} \cdot \overrightarrow{B} = |\overrightarrow{A}||\overrightarrow{B}| \cos(\theta)$
• The dot product can be calculated using components: $\overrightarrow{A} \cdot \overrightarrow{B} = (A_x \hat{i} + A_y \hat{j} + A_z \hat{k})\cdot(B_x \hat{i} + B_y \hat{j} + B_z \hat{k}) = A_xB_x + A_yB_y + A_zB_z$

Vector Product (Cross Product)

• The cross product of two vectors is equal to the product of their magnitudes and the sine of the angle between them, multiplied by a unit vector normal to the plane containing the two vectors: $\overrightarrow{A} \times \overrightarrow{B} = |\overrightarrow{A}||\overrightarrow{B}| \sin(\theta) \hat{n}$
• In component form: $\overrightarrow{A} \times \overrightarrow{B} = (A_x \hat{i} + A_y \hat{j} + A_z \hat{k})\times(B_x \hat{i} + B_y \hat{j} + B_z \hat{k}) = (A_yB_z - A_zB_y)\hat{i} + (A_zB_x - A_xB_z)\hat{j} + (A_xB_y - A_yB_x)\hat{k}$

Mixed Product

• The mixed product of three vectors can be calculated as the determinant of a matrix formed by their components: $\overrightarrow{A} \cdot (\overrightarrow{B} \times \overrightarrow{C}) = \begin{vmatrix} A_x & A_y & A_z \ B_x & B_y & B_z \ C_x & C_y & C_z \end{vmatrix}$

Kinematics

Uniform Rectilinear Motion (MRU)

• Position as a function of time: $x = x_0 + vt$

Uniformly Accelerated Rectilinear Motion (MRUA)

• Position as a function of time: $x = x_0 + v_0t + \frac{1}{2}at^2$
• Velocity as a function of time: $v = v_0 + at$
• Velocity as a function of position: $v^2 = v_0^2 + 2a(x - x_0)$

Projectile Motion

• Horizontal position: $x = x_0 + v_{0x}t$
• Vertical position: $y = y_0 + v_{0y}t - \frac{1}{2}gt^2$
• Horizontal velocity: $v_x = v_{0x}$
• Vertical velocity: $v_y = v_{0y} - gt$

Uniform Circular Motion (MCU)

• Arc length: $s = r\theta$
• Linear velocity: $v = r\omega$
• Linear acceleration: $a = r\alpha$
• Angular velocity: $\omega = \frac{d\theta}{dt}$
• Angular acceleration: $\alpha = \frac{d\omega}{dt}$

Dynamics

Newton's Laws

• 1st Law: Net force is zero implies constant velocity: $\sum \overrightarrow{F} = 0 \Rightarrow \overrightarrow{v} = cte.$
• 2nd Law: Net force equals mass times acceleration: $\sum \overrightarrow{F} = m\overrightarrow{a}$
• 3rd Law: Action-reaction forces are equal and opposite: $\overrightarrow{F}{AB} = -\overrightarrow{F}{BA}$

Work and Energy

• Work done by a force: $W = \overrightarrow{F} \cdot \overrightarrow{d} = |\overrightarrow{F}||\overrightarrow{d}| \cos(\theta)$
• Kinetic energy: $K = \frac{1}{2}mv^2$
• Gravitational potential energy: $U_g = mgh$
• Elastic potential energy: $U_e = \frac{1}{2}kx^2$
• Work-energy theorem: $W = \Delta K + \Delta U$
• Power: $P = \frac{dW}{dt} = \overrightarrow{F} \cdot \overrightarrow{v}$

Impulse and Momentum

• Momentum: $\overrightarrow{p} = m\overrightarrow{v}$
• Impulse: $\overrightarrow{J} = \int \overrightarrow{F} dt = \Delta \overrightarrow{p}$
• Conservation of momentum: $\sum \overrightarrow{p_i} = \sum \overrightarrow{p_f}$

Rotation

• Moment of inertia: $I = \sum m_ir_i^2$
• Kinetic energy of rotation: $K = \frac{1}{2}I\omega^2$
• Torque: $\overrightarrow{\tau} = \overrightarrow{r} \times \overrightarrow{F} = I\overrightarrow{\alpha}$
• Angular momentum: $L = I\omega$

Gravitation

• Gravitational force: $\overrightarrow{F} = -G\frac{m_1m_2}{r^2}\hat{r}$
• Gravitational potential energy: $U = -G\frac{m_1m_2}{r}$

Lecture 19: Lyapunov Stability

Autonomous System

• The system is defined by $\dot{x} = f(x)$, where $x \in \mathbb{R}^n$ and $f: \mathbb{R}^n \rightarrow \mathbb{R}^n$.
• $f$ must be locally Lipschitz, $f(0) = 0$, and $x_e = 0$ is an equilibrium point.

Definition of Stability

• Stable: For every $\epsilon > 0$, there exists $\delta > 0$ such that if $||x(0)|| < \delta$, then $||x(t)|| < \epsilon$ for all $t \geq 0$.
• Unstable: Not stable.
• Asymptotically Stable: Stable, and there exists $\delta > 0$ such that if $||x(0)|| < \delta$, then $\lim_{t \to \infty} x(t) = 0$.
• Globally Asymptotically Stable: Stable, and $\lim_{t \to \infty} x(t) = 0$ for all $x(0) \in \mathbb{R}^n$.

Lyapunov's Direct Method

Lyapunov Function

• $V: \mathbb{R}^n \rightarrow \mathbb{R}$ is a Lyapunov function candidate if:
  • $V(0) = 0$
  • $V(x) > 0$ for all $x \neq 0$
• $V(x)$ must be positive definite.

Theorem

• If there exists a Lyapunov function candidate $V(x)$ such that $\dot{V}(x) \leq 0$ for all $x$, then $x_e = 0$ is stable.
• If $\dot{V}(x) < 0$ for all $x \neq 0$, then $x_e = 0$ is asymptotically stable.
• If $\dot{V}(x) < 0$ for all $x \neq 0$ and $V(x)$ is radially unbounded (i.e., $||x|| \rightarrow \infty \implies V(x) \rightarrow \infty$), then $x_e = 0$ is globally asymptotically stable.

Calculus Definitions

Common Functions

Polynomial

• A polynomial function has the form: $f(x) = a_n x^n + a_{n-1} x^{n-1} +... + a_1 x + a_0$

Rational

• A rational function is a ratio of two polynomials: $f(x) = \frac{P(x)}{Q(x)}$

Algebraic

• Construction uses algebraic operations like addition, multiplication, and roots.

Trigonometric

• Examples: $\sin(x), \cos(x), \tan(x)$ etc.

Exponential

• Exponential functions have the form: $f(x) = a^x$ where a is a positive constant

Logarithmic

• Logarithmic functions have the form: $f(x) = \log_a(x)$ where a is a positive constant

Limit Definition

• $\lim_{x \to a} f(x) = L$ means that $f(x)$ can be made arbitrarily close to L when x is close to a but not equal to a.

Limit Laws

• If $\lim_{x \to a} f(x)$ and $\lim_{x \to a} g(x)$ exist, then:
  • $\lim_{x \to a} [f(x) + g(x)] = \lim_{x \to a} f(x) + \lim_{x \to a} g(x)$
  • $\lim_{x \to a} [f(x) - g(x)] = \lim_{x \to a} f(x) - \lim_{x \to a} g(x)$
  • $\lim_{x \to a} [cf(x)] = c \lim_{x \to a} f(x)$
  • $\lim_{x \to a} [f(x)g(x)] = \lim_{x \to a} f(x) \cdot \lim_{x \to a} g(x)$
  • $\lim_{x \to a} \frac{f(x)}{g(x)} = \frac{\lim_{x \to a} f(x)}{\lim_{x \to a} g(x)}$, if $\lim_{x \to a} g(x) \neq 0$

Continuity

• A function f is continuous at a number a if $\lim_{x \to a} f(x) = f(a)$

Conditions for Continuity

• f must satisfy these conditions to be continuous at a:
  • $f(a)$ is defined
  • $\lim_{x \to a} f(x)$ exists
  • $\lim_{x \to a} f(x) = f(a)$

Types of Discontinuities

• Removable: The limit exists, but $f(a)$ doesn't match the limit or is undefined.
  • Jump: Left and right limits exist but are unequal.
  • Infinite: The function has a vertical asymptote at x = a.

Derivatives

• $f'(x) = \lim_{h \to 0} \frac{f(x+h) - f(x)}{h}$
  • Alternative Definition:
  • $f'(a) = \lim_{x \to a} \frac{f(x) - f(a)}{x - a}$

Basic Rules

Product Rule

• $(fg)' = f'g + fg'$

Quotient Rule

• $(\frac{f}{g})' = \frac{f'g - fg'}{g^2}$

Chain Rule

• $(f(g(x)))' = f'(g(x)) \cdot g'(x)$

Theorems

Intermediate Value Theorem

• If f is continuous on $[a, b]$ and N is between $f(a)$ and $f(b)$, there exists c in $(a, b)$ such that $f(c) = N$.

Mean Value Theorem

• If f is continuous on $[a, b]$ and differentiable on $(a, b)$, there exists c in $(a, b)$ such that $f'(c) = \frac{f(b) - f(a)}{b - a}$.

Static Electricity

Charging

Friction

• Achieved when two neutral materials are rubbed together.
• Objects gain opposite charges.
• One material attracts electrons more strongly.

Conduction

• A charged object makes contact with a neutral object.
• Objects end up with the same charge.

Induction

• A charged object is brought near, but doesn't touch, a neutral object.
• Electrons rearrange within the neutral object (polarization).
• Grounding after polarization results in an opposite charge on the neutral object.

Electric Fields

• A region around a charged object exerts force on other charged objects.
• Electric field lines point away from positives and towards negatives.
• Closer field lines indicate a stronger field.

Electric Potential

• Electric potential is the electric potential energy per unit charge, measured in volts (V).
• Potential difference is voltage.
• Electrons accelerate towards higher potential.

Capacitance

• The ability of a conductor to store energy by separating charge.
• Measured in farads (F).
• Capacitance of a parallel plate capacitor: $C = \epsilon_0 * \frac{A}{d}$, where $\epsilon_0 = 8.85 \times 10^{-12} \frac{C^2}{Nm^2}$, A is plate area, and d is the distance between plates.

Current

• The rate of flow of electric charge, measured in amperes (A).
• Current is given by: $I = \frac{\Delta Q}{\Delta t}$
• One ampere equals one coulomb per second.
• Current flows in the direction of positive charge.
• Electrons flow in the opposite direction.

Resistance

• The opposition to current flow, measured in ohms ($\Omega$).
• $R = \frac{\rho L}{A}$, where $\rho$ is resistivity, L is length, and A is cross-sectional area.

Ohm's Law

• $V = IR$ where V is voltage, I is current, and R is resistance.

Electric Power

• The rate at which electrical energy is converted to other forms.
• Measured in watts (W).
• $P = IV = I^2R = \frac{V^2}{R}$.

Bernoulli's Principle

• States that an increase in fluid speed occurs simultaneously with a decrease in pressure or potential energy.

How Wings Generate Lift

• Airfoil Shape: Curved upper surface, flatter lower surface.
• Airflow Difference: Air travels a longer distance over the curved upper surface.
• Speed and Pressure: Faster air (above) has lower pressure, and slower air (below) has higher pressure.
• Lift Generation: Pressure difference creates an upward force (lift).

Applications

• Airplanes use wings to generate lift.
• Race cars use inverted wings to generate downward force.
• Spray bottles draw fluid up a tube using low pressure.
• Chimneys draw smoke up using low pressure at the top.

Bernoulli's Equation

• $P + \frac{1}{2} \rho v^2 + \rho g h = \text{constant}$
• $P$ is pressure, $\rho$ is density, $v$ is velocity, $g$ is gravity, $h$ is height.
• Applies to ideal, steady-flow fluids.
• Viscosity and turbulence may affect real-world accuracy.