Calculus Formulas for Physics

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1. Basic Derivatives Constant: $\frac{d}{dx}(c) = 0$ Power Rule: $\frac{d}{dx}(x^n) = nx^{n-1}$ Constant Multiple Rule: $\frac{d}{dx}(cf(x)) = c\frac{d}{dx}(f(x))$ Sum/Difference Rule: $\frac{d}{dx}(f(x) \pm g(x)) = \frac{d}{dx}(f(x)) \pm \frac{d}{dx}(g(x))$ Product Rule: $\frac{d}{dx}(f(x)g(x)) = f'(x)g(x) + f(x)g'(x)$ Quotient Rule: $\frac{d}{dx}\left(\frac{f(x)}{g(x)}\right) = \frac{f'(x)g(x) - f(x)g'(x)}{(g(x))^2}$ Chain Rule: $\frac{d}{dx}(f(g(x))) = f'(g(x))g'(x)$ 2. Derivatives of Common Functions Exponential: $\frac{d}{dx}(e^x) = e^x$, $\frac{d}{dx}(a^x) = a^x \ln(a)$ Logarithmic: $\frac{d}{dx}(\ln|x|) = \frac{1}{x}$, $\frac{d}{dx}(\log_a|x|) = \frac{1}{x \ln(a)}$ Trigonometric: $\frac{d}{dx}(\sin x) = \cos x$ $\frac{d}{dx}(\cos x) = -\sin x$ $\frac{d}{dx}(\tan x) = \sec^2 x$ $\frac{d}{dx}(\cot x) = -\csc^2 x$ $\frac{d}{dx}(\sec x) = \sec x \tan x$ $\frac{d}{dx}(\csc x) = -\csc x \cot x$ 3. Basic Integrals (Antiderivatives) Constant: $\int c \, dx = cx + C$ Power Rule: $\int x^n \, dx = \frac{x^{n+1}}{n+1} + C \quad (n \neq -1)$ Logarithmic: $\int \frac{1}{x} \, dx = \ln|x| + C$ Exponential: $\int e^x \, dx = e^x + C$, $\int a^x \, dx = \frac{a^x}{\ln(a)} + C$ Trigonometric: $\int \sin x \, dx = -\cos x + C$ $\int \cos x \, dx = \sin x + C$ $\int \sec^2 x \, dx = \tan x + C$ $\int \csc^2 x \, dx = -\cot x + C$ $\int \sec x \tan x \, dx = \sec x + C$ $\int \csc x \cot x \, dx = -\csc x + C$ 4. Definite Integrals Fundamental Theorem of Calculus: $\int_a^b f(x) \, dx = F(b) - F(a)$ where $F'(x) = f(x)$ Area under a curve: $A = \int_a^b f(x) \, dx$ Average Value: $f_{avg} = \frac{1}{b-a}\int_a^b f(x) \, dx$ 5. Applications in Physics Kinematics Velocity from Position: $v(t) = \frac{dx}{dt}$ Acceleration from Velocity: $a(t) = \frac{dv}{dt} = \frac{d^2x}{dt^2}$ Position from Velocity: $x(t) = \int v(t) \, dt$ Velocity from Acceleration: $v(t) = \int a(t) \, dt$ Displacement: $\Delta x = \int_{t_1}^{t_2} v(t) \, dt$ Work and Energy Work done by a variable force: $W = \int_{x_1}^{x_2} F(x) \, dx$ Power (instantaneous): $P = \frac{dW}{dt} = \vec{F} \cdot \vec{v}$ Moment of Inertia For continuous mass distribution: $I = \int r^2 \, dm$ Center of Mass 1D: $X_{CM} = \frac{\int x \, dm}{\int dm}$ 3D: $\vec{R}_{CM} = \frac{\int \vec{r} \, dm}{\int dm}$ Electric Fields & Potentials Electric field from potential: $\vec{E} = -\nabla V = -\left(\frac{\partial V}{\partial x}\hat{i} + \frac{\partial V}{\partial y}\hat{j} + \frac{\partial V}{\partial z}\hat{k}\right)$ Potential from electric field: $V_B - V_A = -\int_A^B \vec{E} \cdot d\vec{l}$ Electric field of continuous charge distribution: $\vec{E} = \int \frac{k \, dq}{r^2} \hat{r}$ Magnetic Fields Biot-Savart Law: $d\vec{B} = \frac{\mu_0}{4\pi} \frac{I \, d\vec{l} \times \hat{r}}{r^2}$ 6. Multivariable Calculus Basics (for Physics) Partial Derivatives Derivative with respect to one variable, treating others as constants. Example: For $f(x,y) = x^2y + y^3$, $\frac{\partial f}{\partial x} = 2xy$, $\frac{\partial f}{\partial y} = x^2 + 3y^2$ Gradient Vector of partial derivatives: $\nabla f = \left\langle \frac{\partial f}{\partial x}, \frac{\partial f}{\partial y}, \frac{\partial f}{\partial z} \right\rangle$ Points in direction of greatest increase of $f$. Line Integrals For a scalar field $f(x,y,z)$ along curve $C$: $\int_C f \, ds$ For a vector field $\vec{F}$ along curve $C$: $\int_C \vec{F} \cdot d\vec{r}$ Often used for work done by a force: $W = \int_C \vec{F} \cdot d\vec{r}$ Surface Integrals For a scalar field $f(x,y,z)$ over surface $S$: $\iint_S f \, dS$ For a vector field $\vec{F}$ over surface $S$ (Flux): $\iint_S \vec{F} \cdot d\vec{A}$ Gauss's Law: $\oint_S \vec{E} \cdot d\vec{A} = \frac{Q_{enc}}{\epsilon_0}$ Volume Integrals For a scalar field $f(x,y,z)$ over volume $V$: $\iiint_V f \, dV$ Used for total charge, mass, etc. within a volume. Divergence $\nabla \cdot \vec{F} = \frac{\partial F_x}{\partial x} + \frac{\partial F_y}{\partial y} + \frac{\partial F_z}{\partial z}$ Scalar quantity, measures outward flux per unit volume. Divergence Theorem: $\iiint_V (\nabla \cdot \vec{F}) \, dV = \iint_S \vec{F} \cdot d\vec{A}$ Curl $\nabla \times \vec{F} = \left(\frac{\partial F_z}{\partial y} - \frac{\partial F_y}{\partial z}\right)\hat{i} + \left(\frac{\partial F_x}{\partial z} - \frac{\partial F_z}{\partial x}\right)\hat{j} + \left(\frac{\partial F_y}{\partial x} - \frac{\partial F_x}{\partial y}\right)\hat{k}$ Vector quantity, measures rotation. Stokes' Theorem: $\oint_C \vec{F} \cdot d\vec{l} = \iint_S (\nabla \times \vec{F}) \cdot d\vec{A}$