Functions in Vector Calculus

Cheatsheet Content

### Single-Variable Functions ($f: \mathbb{R} \to \mathbb{R}$) - **Definition:** Maps a real number to a real number. $y = f(x)$. - **Examples:** $f(x) = x^2$, $f(x) = \sin(x)$, $f(x) = e^{-x}$. - **Calculus:** Standard derivatives, integrals, limits apply. - $\frac{dy}{dx} = f'(x)$ - $\int f(x) dx$ ### Multivariable Scalar Functions ($f: \mathbb{R}^n \to \mathbb{R}$) - **Definition:** Maps a vector in $\mathbb{R}^n$ to a single real number. $z = f(x, y)$ or $w = f(x, y, z)$. - **Domain:** Typically a region in $\mathbb{R}^2$ or $\mathbb{R}^3$. - **Graph:** For $f(x,y)$, it's a surface in $\mathbb{R}^3$. For $f(x,y,z)$, it's a hypersurface in $\mathbb{R}^4$ (hard to visualize). - **Key Concepts:** - **Partial Derivatives:** $\frac{\partial f}{\partial x}$, $\frac{\partial f}{\partial y}$ - **Gradient:** $\nabla f = \left\langle \frac{\partial f}{\partial x}, \frac{\partial f}{\partial y}, \frac{\partial f}{\partial z} \right\rangle$. Points in the direction of greatest increase. - **Directional Derivative:** $D_{\vec{u}} f = \nabla f \cdot \vec{u}$ - **Level Curves/Surfaces:** $f(x,y) = k$ (curves in $\mathbb{R}^2$), $f(x,y,z) = k$ (surfaces in $\mathbb{R}^3$). - **Multiple Integrals:** $\iint_R f(x,y) dA$, $\iiint_D f(x,y,z) dV$. - **Examples:** - $f(x,y) = x^2 + y^2$ (paraboloid) - $f(x,y,z) = x^2 + y^2 + z^2$ (level surfaces are spheres) ### Vector-Valued Functions of a Single Variable ($\vec{r}: \mathbb{R} \to \mathbb{R}^n$) - **Definition:** Maps a single real number (parameter, often time $t$) to a vector in $\mathbb{R}^n$. - **Notation:** $\vec{r}(t) = \langle x(t), y(t) \rangle$ or $\vec{r}(t) = \langle x(t), y(t), z(t) \rangle$. - **Graph:** A curve in $\mathbb{R}^2$ or $\mathbb{R}^3$. - **Key Concepts:** - **Velocity Vector:** $\vec{v}(t) = \vec{r}'(t) = \langle x'(t), y'(t), z'(t) \rangle$. Tangent to the curve. - **Acceleration Vector:** $\vec{a}(t) = \vec{r}''(t) = \langle x''(t), y''(t), z''(t) \rangle$. - **Speed:** $|\vec{v}(t)| = |\vec{r}'(t)|$. - **Arc Length:** $L = \int_a^b |\vec{r}'(t)| dt$. - **Unit Tangent Vector:** $\vec{T}(t) = \frac{\vec{r}'(t)}{|\vec{r}'(t)|}$. - **Curvature:** $\kappa = \frac{|\vec{T}'(t)|}{|\vec{r}'(t)|} = \frac{|\vec{r}'(t) \times \vec{r}''(t)|}{|\vec{r}'(t)|^3}$. - **Examples:** - $\vec{r}(t) = \langle \cos t, \sin t \rangle$ (circle in $\mathbb{R}^2$) - $\vec{r}(t) = \langle \cos t, \sin t, t \rangle$ (helix in $\mathbb{R}^3$) ### Vector Fields ($\vec{F}: \mathbb{R}^n \to \mathbb{R}^n$) - **Definition:** Maps a vector in $\mathbb{R}^n$ to another vector in $\mathbb{R}^n$. - **Notation:** - $\vec{F}(x,y) = \langle P(x,y), Q(x,y) \rangle$ (2D vector field) - $\vec{F}(x,y,z) = \langle P(x,y,z), Q(x,y,z), R(x,y,z) \rangle$ (3D vector field) - **Visualization:** At each point, an arrow representing the vector at that point. - **Key Concepts:** - **Line Integrals:** $\int_C \vec{F} \cdot d\vec{r}$ (work done by a force field). - **Conservative Vector Field:** If $\vec{F} = \nabla f$ for some scalar function $f$ (potential function). - Test: $\frac{\partial Q}{\partial x} = \frac{\partial P}{\partial y}$ (2D), $\nabla \times \vec{F} = \vec{0}$ (3D). - **Curl:** $\nabla \times \vec{F}$. Measures rotation. - **Divergence:** $\nabla \cdot \vec{F}$. Measures expansion/contraction. - **Green's Theorem:** Relates line integral to double integral. - **Stokes' Theorem:** Relates surface integral of curl to line integral. - **Divergence Theorem:** Relates triple integral of divergence to surface integral. - **Examples:** - $\vec{F}(x,y) = \langle -y, x \rangle$ (rotational field) - $\vec{F}(x,y,z) = \langle x, y, z \rangle$ (radial field)