Vector Calculus Topic Summaries PDF

# Cylindrical and Spherical Coordinates ## Cylindrical Coordinates The position vector of a point in cylindrical coordinates is given by: - $x = (\rho cos \theta, \rho sin \theta, z)$ ## Spherical Coordinates The position vector of a point in spherical coordinates is given by: - $x = (\rho cos...

# Cylindrical and Spherical Coordinates ## Cylindrical Coordinates The position vector of a point in cylindrical coordinates is given by: - $x = (\rho cos \theta, \rho sin \theta, z)$ ## Spherical Coordinates The position vector of a point in spherical coordinates is given by: - $x = (\rho cos \theta sin \phi, \rho sin \theta sin \phi, \rho cos \phi)$ ## Change of Variables - $\iint_R f(x, y) dx dy = \iint_S f(x(u, v), y(u, v)) |\frac{\partial(x, y)}{\partial(u, v)}| du dv $ - $\frac{\partial(x, y)}{\partial(u, v)} = \begin{vmatrix} \frac{\partial x}{\partial u} & \frac{\partial x}{\partial v} \\ \frac{\partial y}{\partial u} & \frac{\partial y}{\partial v} \end{vmatrix} $ ## 3D: Cylindrical - $\frac{\partial(x, y, z)}{\partial(\rho, \theta, z)} = \begin{vmatrix} \frac{\partial x}{\partial \rho} & \frac{\partial x}{\partial \theta} & \frac{\partial x}{\partial z} \\ \frac{\partial y}{\partial \rho} & \frac{\partial y}{\partial \theta} & \frac{\partial y}{\partial z} \\ \frac{\partial z}{\partial \rho} & \frac{\partial z}{\partial \theta} & \frac{\partial z}{\partial z} \end{vmatrix} = \begin{vmatrix} cos\theta & - \rho sin \theta & 0 \\ sin\theta & \rho cos\theta & 0 \\ 0 & 0 & 1 \end{vmatrix} = \rho $ - $\iiint_R f(x, y, z) dx dy dz = \iiint_S g(\rho, \theta, z) \rho d\rho d\theta dz $ ## Spherical - $\frac{\partial(x, y, z)}{\partial(\rho, \theta, \phi)} = \begin{vmatrix} \frac{\partial x}{\partial \rho} & \frac{\partial x}{\partial \theta} & \frac{\partial x}{\partial \phi} \\ \frac{\partial y}{\partial \rho} & \frac{\partial y}{\partial \theta} & \frac{\partial y}{\partial \phi} \\ \frac{\partial z}{\partial \rho} & \frac{\partial z}{\partial \theta} & \frac{\partial z}{\partial \phi} \end{vmatrix} = \begin{vmatrix} cos\theta sin\phi & - \rho sin\theta sin\phi & \rho cos\theta cos\phi \\ sin\theta sin\phi & \rho cos\theta sin\phi & \rho sin\theta cos\phi \\ cos\phi & 0 & - \rho sin\phi \end{vmatrix} = \rho^2 sin\phi$ - $\iiint_R f(x, y, z) dx dy dz = \iiint_S g(\rho, \theta, \phi) \rho^2 sin \phi d\rho d\theta d\phi $ # Problem Sheet 1, Q4 Find the region bounded above by the sphere $x^2 + y^2 + z^2 = a^2$ and below by the cone $z^2 sin^2\alpha = (x^2 + y^2) cos^2 \alpha$, where $0 < \alpha < \pi$ is a constant. - $x^2 + y^2 + z^2 = a^2$ - $\implies \rho^2 = a^2$ - $\implies \rho = a$ - $z^2 sin^2\alpha = (x^2 + y^2) cos^2\alpha$ - $\implies z^2 = (x^2 + y^2) cot^2\alpha$ - $\implies z = \rho cos\phi = \rho sin\phi cot\alpha$ - $\implies tan \phi = tan \alpha$ - $\implies \phi = \alpha$ Therefore, the volume of the region is given by: - $V = \int_0^{2\pi} \int_0^\alpha \int_0^a \rho^2 sin\phi d\rho d\phi d\theta$ - $V = \frac{2}{3} a^3 (1 - cos \alpha)$ - $V = \frac{1}{3} a^3 (\alpha - sin 2\alpha)$ # 3. Directional Derivatives, Grad. & Potentials ## Gradient: - Let $f: \mathbb{R}^3 \to \mathbb{R}, f: \mathbb{R}^3 \to \mathbb{R}$ be a scalar field. - $\nabla f = \frac{\partial f}{\partial x}\hat{i} + \frac{\partial f}{\partial y} \hat{j} + \frac{\partial f}{\partial z} \hat{k}$ - $\nabla(\lambda f + \mu g) = \lambda \nabla f + \mu \nabla g $ - $\nabla(f\cdot g) = f \nabla g + g\nabla f $ ## Chain Rule: - Let $f$ be a scalar field. - $\frac{df}{dt} = \frac{\partial f}{\partial x} \frac{\partial x}{\partial t} + \frac{\partial f}{\partial y} \frac{\partial y}{\partial t} + \frac{\partial f}{\partial z} \frac{\partial z}{\partial t} $ ## Directional Derivative - $D_a f(x) = \nabla f(x) \cdot a = |\nabla f(x)| cos\theta$ - $\theta$ is the angle between $\nabla f(x)$ and $a$. ## Conservative Vector Field - $F = \nabla \phi$ - $\phi$ is a scalar potential. # 4. Curves & Line Integrals ## Curve in $R^3$ - Let $C \subset \mathbb{R}^3$ with parameterization $\gamma: [a, b] \to \mathbb{R}^3$ such that - $[\gamma(a), \gamma(b)] = C$ - $\gamma$ is continuous. - $\gamma(a)$ is the start point and $\gamma(b)$ is the end point. - A simple curve does not intersect itself. - A closed curve has the property that $\gamma(a) = \gamma(b)$ ## Tangent Vector - The tangent vector at a point $\gamma(t)$ on the curve is given by $\frac{d\gamma}{dt}$. ## Arc Length - $s(t) = \int_{a}^{t} |\frac{d\gamma}{dt'}| dt'$ ## Line Integral of a Scalar Field - $\int_C f ds := \int_{a}^{b} f(\gamma(t)) |\frac{d\gamma}{dt}| dt$ ## Line Integral of a Vector Field - $\int_{C} F \cdot dr = \int_{a}^{b} F(\gamma(t)) \cdot \frac{d\gamma}{dt} dt$ ## Example: Line Integral of a Scalar Field Find the line integral of $f(x, y, z) = x + y + z$ where $C$ is the straight line between the points $(1, 2, 3)$ and $(4, 5, 6)$. - $\gamma(t) = a + t(b - a) = (1, 2, 3) + t((4, 5, 6) - (1, 2, 3)) = (1, 2, 3) + t(3, 3, 3) = (1 + 3t, 2 + 3t, 3 + 3t)$ - $f(\gamma(t)) = (1 + 3t) + (2 + 3t) + (3 + 3t) = 6 + 9t$ - $|\frac{d\gamma}{dt}| = \sqrt{3^2 + 3^2 + 3^2} = 3 \sqrt{3}$ - $\int_C f ds = \int_0^1 (6 + 9t) 3 \sqrt{3} dt = 63 \sqrt{3}$ # 5. Surfaces ## Representation | Representation| Definition| Unit Normal Vector| Surface Element | |---|---|---|---| | Explicit | $z = f(x, y)$ | $\hat{n} = \frac{(\frac{-\partial f}{\partial x}, \frac{-\partial f}{\partial y}, 1)}{\sqrt{(\frac{\partial f}{\partial x})^2 + (\frac{\partial f}{\partial y})^2 + 1}}$ | $dS = \sqrt{1 + (\frac{\partial f}{\partial x})^2 + (\frac{\partial f}{\partial y})^2} dx dy $ | | Implicit | $F(x, y, z) = 0$ | $\hat{n} = \frac{\nabla F}{|\nabla F|}$ | $dS = \frac{|\nabla F|}{|\frac{\partial F}{\partial z}|} dx dy $ | | Parametric | $\mathbf{r}(u, v) = (x(u, v), y(u, v), z(u, v))$ | $\hat{n} = \frac{\frac{\partial \mathbf{r}}{\partial u} \times \frac{\partial \mathbf{r}}{\partial v}}{|\frac{\partial \mathbf{r}}{\partial u} \times \frac{\partial \mathbf{r}}{\partial v}|}$ | $dS = |\frac{\partial \mathbf{r}}{\partial u} \times \frac{\partial \mathbf{r}}{\partial v} | du dv $ | # 6. Divergence & Curl ## Jacobian Matrix - Let $f: \mathbb{R}^m \to \mathbb{R}^n$. - The jacobian matrix of $f$ is $Df(x) = \begin{pmatrix} \frac{\partial f_1}{\partial x_1} & \dots & \frac{\partial f_1}{\partial x_m} \\ \vdots & \ddots & \vdots \\ \frac{\partial f_n}{\partial x_1} & \dots & \frac{\partial f_n}{\partial x_m} \end{pmatrix}$ ## Divergence - Let $D \subset \mathbb{R}^3$ and $F : D \to \mathbb{R}^3$ be a vector field. - $div F = \nabla \cdot F = \frac{\partial F_1}{\partial x} + \frac{\partial F_2}{\partial y} + \frac{\partial F_3}{\partial z}$ - $div F$ measures the expansion of the vector field at a point. - Given a point, the vector is the vector from the origin to that point, but put on that point. ## Properties of Divergence - $\nabla \cdot (\lambda F + \mu G) = \lambda \nabla \cdot F + \mu \nabla \cdot G$ - $\nabla \cdot (fE) = (\nabla f) \cdot F + f (\nabla \cdot F)$ ## Solenoidal/Incompressible - A vector field is solenoidal or incompressible if $\nabla \cdot F = 0$. ## Curl - Let $D \subset \mathbb{R}^3$ and $F : D \to \mathbb{R}^3$ be a vector field. - $curl F = \nabla \times F = \begin{vmatrix} \hat{i} & \hat{j} & \hat{k} \\ \frac{\partial}{\partial x} & \frac{\partial}{\partial y} & \frac{\partial}{\partial z} \\ F_1 & F_2 & F_3 \end{vmatrix} = \Big( \frac{\partial F_3}{\partial y} - \frac{\partial F_2}{\partial z} \Big)\hat{i} - \Big( \frac{\partial F_3}{\partial x} - \frac{\partial F_1}{\partial z} \Big)\hat{j} + \Big( \frac{\partial F_2}{\partial x} - \frac{\partial F_1}{\partial y} \Big)\hat{k} $ - $curl F$ measures the rotation of the vector field at a point. - Given a point, the vector is the vector from the origin to that point, but put on that point. ## Properties of Curl - $\nabla \times (\lambda F + \mu G) = \lambda \nabla \times F + \mu \nabla \times G$ - $\nabla \times (fE) = (\nabla f) \times F + f (\nabla \times F)$ ## Irrotational - A vector field is irrotational if $\nabla \times F = 0$. ## Second Derivatives for Vector Field - $grad(div F) = \nabla(\nabla \cdot F)$ - $div(curl F) = \nabla \cdot (\nabla \times F)$ - $curl(curl F) = \nabla \times (\nabla \times F)$ ## Laplace Operator - For a scalar field, the Laplace operator is given by $\nabla^2 f = \nabla \cdot (\nabla f)$. - For a vector field, the Laplace operator is given by $\nabla^2 F = \begin{pmatrix} \nabla^2 F_1 \\ \nabla^2 F_2 \\ \nabla^2 F_3 \end{pmatrix}$. ## Laplace's Equation - $\nabla^2 u = 0$ ## Solenoidal and Irrotational - If $u$ is a solution to Laplace's equation, then $\nabla u$ is both solenoidal and irrotational. # 7. Higher Dimension of Fundamental Theorem of Calculus ## Geometric Definitions | Name| Definition| Diagram| |---|---|---| | Domain | Open, connected subset of $\mathbb{R}^3$ | [Diagram of a domain] | | Bounded | $\mathbb{R}^3 \backslash O$ such that $\mathbb{R}^3 \backslash O$ is a subset of $\mathbb{R}^3$ where $B_R = \{x : |x| < R\}$ is a ball with radius $R$ and center $O$| [Diagram of a ball] | | Convex| The line connecting two points $x_1, x_2 \in D$ is a subset of $D$ | [Diagram of a convex domain] | | Open | For any two points $x_1, x_2 \in S$, there exists a curve from $x_1$ to $x_2$ not crossing S. | [Diagram of an open surface ] | | Closed | Not open | [Diagram of a closed surface] | # 8. Green's Theorem - Let $D \subset \mathbb{R}^2$ be a bounded domain with boundary curve $C$. - $C$ is a simple, sufficiently smooth, and oriented anticlockwise curve. - Let $F$ be a vector field. - $\oint_C F \cdot dr = \iint_D (\frac{\partial F_2}{\partial x} - \frac{\partial F_1}{\partial y}) dA$ ## Explanation of Green's Theorem Green's theorem is a result that relates the line integral of a vector field around a simple closed curve to the double integral of the curl of the vector field over the region enclosed by the curve. # 9. Divergence Theorem Let $D \subset \mathbb{R}^3$ be a bounded domain with boundary $\partial D$. Let $\Delta$ be the outward-pointing unit normal vector on $\partial D$. Let $F$ be a sufficiently nice vector field on $D$. - $\iint_{\partial D} F \cdot \Delta dS =\iiint_D \nabla \cdot F dV$ ## Explanation of the Divergence Theorem The divergence theorem is a generalization of Green's theorem to three dimensions. It states that the flux of a vector field across a closed surface is equal to the integral of the divergence of the vector field over the volume enclosed by the surface. # 10. Stokes' Theorem - Let $S$ be a sufficiently smooth, bounded, open, orientable surface in $\mathbb{R}^3$ with outward-pointing unit normal vector $\Delta$. - Let $C$ be its closed boundary curve. - Let $F$ be a sufficiently nice vector field in a neighborhood of $S$. - $\oint_C F \cdot dr = \iint_S (\nabla \times F) \cdot \Lambda dS$ ## Explanation of Stokes' Theorem Stokes' theorem relates the line integral of a vector field around a closed curve to the flux of the curl of the vector field across any surface that is bounded by the curve. ## Connection between Divergence and Stokes' Theorems Both divergence and Stokes' theorem are fundamental theorems of vector calculus that relate integrals over volumes or surfaces to integrals over their boundaries. The divergence theorem relates the flux of a vector field across a closed surface to the integral of the divergence of the vector field over the volume enclosed by the surface. Stokes' theorem relates the line integral of a vector field around a closed curve to the flux of the curl of the vector field across any surface that is bounded by the curve. The two theorems are both generalizations of the fundamental theorem of calculus to higher dimensions.