Elementary Analysis III Cheats

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### 1. Functions of More Than One Variable #### 1.1 Introduction - **Definition 1.1.1: Function of Two Variables** A function $f$ of two variables $x$ and $y$ is a correspondence from a set $D \subseteq \mathbb{R}^2$ to $\mathbb{R}$ such that each $(x, y) \in D$ is associated with a unique real number $f(x, y)$. - **Definition 1.1.2: Function of $n$ Variables** A function $f$ of $n$ variables $x_1, \dots, x_n$ is a correspondence from a set $D \subseteq \mathbb{R}^n$ to $\mathbb{R}$ such that each $(x_1, \dots, x_n) \in D$ is associated with a unique real number $f(x_1, \dots, x_n)$. - $D$ is the **domain** of $f$ (denoted $\text{dom } f$). - The set of all resulting values $f(x_1, \dots, x_n)$ is the **range** of $f$ (denoted $\text{ran } f$). - **Definition 1.1.3: Equality of Functions** Two functions $f: D_1 \to \mathbb{R}$ and $g: D_2 \to \mathbb{R}$ are equal iff $D_1 = D_2$ and $f(x_1, \dots, x_n) = g(x_1, \dots, x_n)$ for all $(x_1, \dots, x_n) \in D_1$. - **Graphs of Functions of Two Variables:** The graph of $f(x, y)$ is the set of all points $(x, y, z)$ in $\mathbb{R}^3$ such that $(x, y) \in \text{dom } f$ and $z = f(x, y)$. This forms a surface in $\mathbb{R}^3$. - **Level Curves:** For a function $f(x, y)$, a level curve is the set of points $(x, y)$ where $f(x, y) = k$ for some constant $k$. These are projections of horizontal cross-sections of the graph $z = f(x, y)$ onto the $xy$-plane. - **Level Surfaces:** For a function $f(x, y, z)$, a level surface is the set of points $(x, y, z)$ where $f(x, y, z) = k$ for some constant $k$. Useful for visualizing functions of three variables. #### 1.2 Limits and Continuity - **Definition 1.2.1: Limit of a Function of Two Variables** $\lim_{(x,y) \to (a,b)} f(x, y) = L$ if for every $\epsilon > 0$, there is a $\delta > 0$ such that $|f(x, y) - L| #### 1.3 Partial Derivatives - **Definition 1.3.1: Partial Derivatives** For $z = f(x, y)$: 1. **With respect to $x$**: $f_x(x, y) = \lim_{\Delta x \to 0} \frac{f(x + \Delta x, y) - f(x, y)}{\Delta x}$ (treat $y$ as constant). 2. **With respect to $y$**: $f_y(x, y) = \lim_{\Delta y \to 0} \frac{f(x, y + \Delta y) - f(x, y)}{\Delta y}$ (treat $x$ as constant). - **Notations:** $f_x(x, y)$, $D_x f(x, y)$, $\frac{\partial}{\partial x} f(x, y)$, $\frac{\partial z}{\partial x}$. - **Higher Order Partial Derivatives:** - $f_{xx} = \frac{\partial^2 z}{\partial x^2}$ - $f_{xy} = \frac{\partial^2 z}{\partial y \partial x}$ (differentiate with respect to $x$ first, then $y$) - $f_{yx} = \frac{\partial^2 z}{\partial x \partial y}$ (differentiate with respect to $y$ first, then $x$) - $f_{yy} = \frac{\partial^2 z}{\partial y^2}$ - **Theorem 1.3.2: Clairaut's Theorem (Equality of Mixed Partials)** If $f_{xy}$ and $f_{yx}$ are continuous on some circular region $D$, then $f_{xy} = f_{yx}$ on $D$. #### 1.4 Differentiability, Differentials and Local Linear Approximation - **Definition 1.4.1: Differentiability** A function $f(x, y)$ is differentiable at $(x_0, y_0)$ if $f_x(x_0, y_0)$ and $f_y(x_0, y_0)$ both exist and $\lim_{(\Delta x, \Delta y) \to (0,0)} \frac{\Delta f - f_x(x_0, y_0)\Delta x - f_y(x_0, y_0)\Delta y}{\sqrt{(\Delta x)^2 + (\Delta y)^2}} = 0$, where $\Delta f = f(x_0 + \Delta x, y_0 + \Delta y) - f(x_0, y_0)$. - **Theorem 1.4.2:** 1. If a function is differentiable at a point, then it is continuous at that point. 2. If all first-order partial derivatives of $f$ exist and are continuous at a point, then $f$ is differentiable at that point. - **Definition 1.4.3: Total Differential** For $z = f(x, y)$ differentiable at $(x_0, y_0)$, the total differential of $z$ is $dz = f_x(x_0, y_0) dx + f_y(x_0, y_0) dy$. - It approximates the change in $f$, $\Delta f \approx dz$. - For $n$ variables: $du = \frac{\partial u}{\partial x_1} dx_1 + \dots + \frac{\partial u}{\partial x_n} dx_n$. - **Definition 1.4.4: Local Linear Approximation (Linearization)** For $f(x, y)$ differentiable at $(x_0, y_0)$, the local linear approximation is $L(x, y) = f(x_0, y_0) + f_x(x_0, y_0)(x - x_0) + f_y(x_0, y_0)(y - y_0)$. - This is the equation of the **tangent plane** to the surface $z = f(x, y)$ at $(x_0, y_0)$. #### 1.5 Chain Rule - **Theorem 1.5.1: General Chain Rule** If $u = f(x_1, \dots, x_n)$ is differentiable, and each $x_i = g_i(t_1, \dots, t_m)$ is differentiable, then $u$ is a differentiable function of $t_1, \dots, t_m$. For each $j = 1, \dots, m$: $\frac{\partial u}{\partial t_j} = \frac{\partial u}{\partial x_1}\frac{\partial x_1}{\partial t_j} + \frac{\partial u}{\partial x_2}\frac{\partial x_2}{\partial t_j} + \dots + \frac{\partial u}{\partial x_n}\frac{\partial x_n}{\partial t_j}$. - **Special Case: $z = f(x, y)$, $x = x(t)$, $y = y(t)$** $\frac{dz}{dt} = \frac{\partial z}{\partial x}\frac{dx}{dt} + \frac{\partial z}{\partial y}\frac{dy}{dt}$. - **Special Case: $z = f(x, y)$, $x = x(s, t)$, $y = y(s, t)$** $\frac{\partial z}{\partial s} = \frac{\partial z}{\partial x}\frac{\partial x}{\partial s} + \frac{\partial z}{\partial y}\frac{\partial y}{\partial s}$ $\frac{\partial z}{\partial t} = \frac{\partial z}{\partial x}\frac{\partial x}{\partial t} + \frac{\partial z}{\partial y}\frac{\partial y}{\partial t}$. - **Theorem 1.5.2: Implicit Differentiation for $F(x, y) = 0$** If $y$ is a differentiable function of $x$ defined by $F(x, y) = 0$, then $\frac{dy}{dx} = -\frac{F_x(x, y)}{F_y(x, y)}$ (provided $F_y \ne 0$). - **Theorem 1.5.3: Implicit Differentiation for $F(x, y, z) = 0$} If $z$ is a differentiable function of $x$ and $y$ defined by $F(x, y, z) = 0$, then $\frac{\partial z}{\partial x} = -\frac{F_x}{F_z}$ and $\frac{\partial z}{\partial y} = -\frac{F_y}{F_z}$ (provided $F_z \ne 0$). #### 1.6 Directional Derivatives and Gradients - **Definition 1.6.1: Directional Derivative** For $f(x, y)$ and unit vector $\vec{u} = \langle u_1, u_2 \rangle$: $D_{\vec{u}}f(x, y) = \lim_{h \to 0} \frac{f(x + hu_1, y + hu_2) - f(x, y)}{h}$. - $D_{\vec{u}}f(x_0, y_0)$ is the rate of change of $f$ along $\vec{u}$ at $(x_0, y_0)$. - **Definition 1.6.2: Gradient** For $f(x, y)$ with existing $f_x, f_y$: $\vec{\nabla}f(x, y) = \langle f_x(x, y), f_y(x, y) \rangle$. - For $f(x, y, z)$: $\vec{\nabla}f(x, y, z) = \langle f_x(x, y, z), f_y(x, y, z), f_z(x, y, z) \rangle$. - **Theorem 1.6.3:** If $f$ is differentiable, $D_{\vec{u}}f(x, y) = \vec{\nabla}f(x, y) \cdot \vec{u}$. - **Theorem 1.6.4: Max/Min Rate of Change** 1. Maximum rate of change of $f$ at a point is $\|\vec{\nabla}f\|$ and occurs in the direction of $\vec{\nabla}f$. 2. Minimum rate of change of $f$ at a point is $-\|\vec{\nabla}f\|$ and occurs in the direction of $-\vec{\nabla}f$. - **Theorem 1.6.5: Gradient Normal to Level Surface** If $S$ is a surface $F(x, y, z) = k$, then $\vec{\nabla}F(x_0, y_0, z_0)$ is normal to $S$ at $(x_0, y_0, z_0)$. - **Tangent Plane:** To $F(x, y, z) = k$ at $(x_0, y_0, z_0)$ is $\vec{\nabla}F(x_0, y_0, z_0) \cdot \langle x-x_0, y-y_0, z-z_0 \rangle = 0$. - **Normal Line:** To $F(x, y, z) = k$ at $(x_0, y_0, z_0)$ is $\vec{R}(t) = \langle x_0, y_0, z_0 \rangle + t \vec{\nabla}F(x_0, y_0, z_0)$. - **Theorem 1.6.6: Gradient Orthogonal to Level Curve** If $C$ is a curve $f(x, y) = k$, then $\vec{\nabla}f(x_0, y_0)$ is orthogonal to $C$ at $(x_0, y_0)$. #### 1.7 Relative Extrema of Functions of Two Variables - **Definition 1.7.1: Relative Extrema** 1. $f$ has a **relative maximum** at $(x_0, y_0)$ if $f(x_0, y_0) \ge f(x, y)$ for all $(x, y)$ in some disk centered at $(x_0, y_0)$. 2. $f$ has a **relative minimum** at $(x_0, y_0)$ if $f(x_0, y_0) \le f(x, y)$ for all $(x, y)$ in some disk centered at $(x_0, y_0)$. - **Theorem 1.7.2:** If $f$ has a relative extremum at $(x_0, y_0)$ and $f_x, f_y$ exist at $(x_0, y_0)$, then $f_x(x_0, y_0) = 0$ and $f_y(x_0, y_0) = 0$. - **Definition 1.7.3: Critical Point** $(x_0, y_0)$ is a critical point if: 1. $f_x(x_0, y_0) = 0$ and $f_y(x_0, y_0) = 0$, OR 2. $f_x(x_0, y_0)$ and/or $f_y(x_0, y_0)$ does not exist. - **Definition 1.7.4: Saddle Point** A critical point $(x_0, y_0)$ is a saddle point if $f$ does not have a relative extremum there. - **Theorem 1.7.5: Second Derivatives Test** Let $\vec{\nabla}f(x_0, y_0) = \vec{0}$ and $f_{xx}, f_{yy}, f_{xy}$ be continuous near $(x_0, y_0)$. Let $D(x_0, y_0) = f_{xx}(x_0, y_0)f_{yy}(x_0, y_0) - (f_{xy}(x_0, y_0))^2$. 1. If $D > 0$ and $f_{xx} > 0$, then $f$ has a relative minimum at $(x_0, y_0)$. 2. If $D > 0$ and $f_{xx} #### 1.8 Absolute Extrema of Functions of More than One Variable - **Definition 1.8.1: Absolute Extrema** 1. $f$ has an **absolute maximum** at $(x_0, y_0)$ if $f(x_0, y_0) \ge f(x, y)$ for all $(x, y) \in D$. 2. $f$ has an **absolute minimum** at $(x_0, y_0)$ if $f(x_0, y_0) \le f(x, y)$ for all $(x, y) \in D$. - **Definition 1.8.3: Closed and Bounded Regions** 1. A region $D$ is **closed** if it contains all its boundary points. 2. A region $D$ is **bounded** if it is a subregion of a closed disk. - **Theorem 1.8.4: Extreme Value Theorem for Functions of Two Variables** If $f$ is continuous on a closed and bounded region $D$, then $f$ has an absolute maximum and an absolute minimum on $D$. - **Steps to find Absolute Extrema on a Closed and Bounded Region $D$:** 1. Find all critical points of $f$ inside $D$. 2. Find extrema of $f$ on the boundary of $D$ (often by parametrizing boundary segments and reducing to single-variable optimization). 3. Evaluate $f$ at all points found in steps 1 and 2, and at the intersection points of boundary curves. The largest value is the absolute maximum, and the smallest is the absolute minimum. - **Theorem 1.8.5: Lagrange Multipliers** To find extrema of $f(x, y)$ subject to $g(x, y) = k$: if $f$ has a relative extremum at $(x_0, y_0)$ subject to $g(x, y) = k$ and $\vec{\nabla}g(x_0, y_0) \ne \vec{0}$, then there exists a constant $\lambda$ such that $\vec{\nabla}f(x_0, y_0) = \lambda \vec{\nabla}g(x_0, y_0)$. - **Method of Lagrange Multipliers:** - For $f(x, y)$ subject to $g(x, y) = k$, solve the system: $f_x = \lambda g_x$ $f_y = \lambda g_y$ $g(x, y) = k$ - For $f(x, y, z)$ subject to $g(x, y, z) = k$, solve the system: $f_x = \lambda g_x$ $f_y = \lambda g_y$ $f_z = \lambda g_z$ $g(x, y, z) = k$ - Evaluate $f$ at all solutions. The largest is the max, smallest is the min. #### 1.9 Parametric Surfaces - **Parametric Surface Definition:** A surface $S$ can be represented by a vector-valued function $\vec{R}(u, v) = \langle x(u, v), y(u, v), z(u, v) \rangle$ for $(u, v)$ in a region $D$ in the $uv$-plane. - $x = x(u, v)$, $y = y(u, v)$, $z = z(u, v)$ are its parametric equations. - $u, v$ are parameters. - **Constant u-curves / v-curves:** Holding one parameter constant (e.g., $u=u_0$) and varying the other generates curves on the surface. - **Surfaces of Revolution:** - Revolving $y = f(x)$ about the $x$-axis: $\vec{R}(x, \theta) = \langle x, f(x)\cos\theta, f(x)\sin\theta \rangle$. - Revolving $x = f(y)$ about the $y$-axis: $\vec{R}(y, \theta) = \langle f(y)\cos\theta, y, f(y)\sin\theta \rangle$. - Revolving $y = f(z)$ about the $z$-axis: $\vec{R}(z, \theta) = \langle f(z)\sin\theta, f(z)\cos\theta, z \rangle$. - **Partial Derivatives of $\vec{R}$:** $\vec{R}_u(u, v) = \langle x_u, y_u, z_u \rangle$ $\vec{R}_v(u, v) = \langle x_v, y_v, z_v \rangle$ - **Theorem 1.9.2: Normal Vector to Parametric Surface** If $\vec{R}_u(u_0, v_0)$ and $\vec{R}_v(u_0, v_0)$ exist and are nonzero, then $\vec{R}_u(u_0, v_0) \times \vec{R}_v(u_0, v_0)$ is normal to $S$ at the point corresponding to $(u_0, v_0)$. - This vector can be used to find the equation of the tangent plane at a point. ### 2. Multiple Integration #### 2.1 Double Integrals over Rectangular Regions - **Motivation: Volume Problem** Find the volume of the solid under $z = f(x, y)$ over a rectangular region $R = [a, b] \times [c, d]$, where $f(x, y) \ge 0$. - **Riemann Sum for Double Integrals:** Partition $R$ into $N$ subrectangles $R_k$ with area $\Delta A_k$. Choose a sample point $(x_k^*, y_k^*) \in R_k$. The volume is approximated by $\sum_{k=1}^N f(x_k^*, y_k^*)\Delta A_k$. - **Definition 2.1.1: Double Integral** $\iint_R f(x, y) \, dA = \lim_{\|P\| \to 0} \sum_{k=1}^N f(x_k^*, y_k^*)\Delta A_k$, if the limit exists. - **Theorem 2.1.2:** If $f$ is continuous on a closed rectangular region $R$, then $f$ is integrable on $R$. - **Theorem 2.1.3: Fubini's Theorem (for Rectangular Regions)** If $f$ is continuous on $R = [a, b] \times [c, d]$, then: $\iint_R f(x, y) \, dA = \int_c^d \int_a^b f(x, y) \, dx \, dy = \int_a^b \int_c^d f(x, y) \, dy \, dx$. - **Iterated Integrals:** Evaluate from the inside out. For $\int_c^d \int_a^b f(x, y) \, dx \, dy$, integrate with respect to $x$ first (treating $y$ as constant), then integrate the result with respect to $y$. #### 2.2 Double Integrals Over General Regions - **Definition 2.2.1: Double Integral over General Region** For a continuous function $f$ on a closed and bounded region $R$, define $F(x, y) = f(x, y)$ if $(x, y) \in R$ and $0$ otherwise. Then $\iint_R f(x, y) \, dA = \iint_D F(x, y) \, dA$ where $D$ is a rectangle enclosing $R$. - Can be interpreted as the net signed volume. - **Type I Region:** $R = \{(x, y) \mid a \le x \le b, g_1(x) \le y \le g_2(x)\}$. - **Theorem 2.2.2:** $\iint_R f(x, y) \, dA = \int_a^b \int_{g_1(x)}^{g_2(x)} f(x, y) \, dy \, dx$. (Integrate along vertical strips). - **Type II Region:** $R = \{(x, y) \mid h_1(y) \le x \le h_2(y), c \le y \le d\}$. - **Theorem 2.2.3:** $\iint_R f(x, y) \, dA = \int_c^d \int_{h_1(y)}^{h_2(y)} f(x, y) \, dx \, dy$. (Integrate along horizontal strips). - **Properties of Double Integrals:** 1. $\iint_R c f(x, y) \, dA = c \iint_R f(x, y) \, dA$. 2. $\iint_R [f(x, y) \pm g(x, y)] \, dA = \iint_R f(x, y) \, dA \pm \iint_R g(x, y) \, dA$. 3. If $R = R_1 \cup R_2$ (non-overlapping), $\iint_R f(x, y) \, dA = \iint_{R_1} f(x, y) \, dA + \iint_{R_2} f(x, y) \, dA$. - **Reversing the Order of Integration:** Sometimes necessary to simplify evaluation. Sketch the region of integration to identify new limits. #### 2.3 Double Integrals in Polar Coordinates - **Coordinate Transformation:** $x = r \cos\theta$, $y = r \sin\theta$, $r^2 = x^2 + y^2$, $\tan\theta = y/x$. - **Polar Rectangle:** Region enclosed by $r=a, r=b, \theta=\alpha, \theta=\beta$. - **Area Element in Polar Coordinates:** $dA = r \, dr \, d\theta$. - **Theorem 2.3.1: Double Integral over Polar Rectangle** If $f$ is continuous on $R = \{(r, \theta) \mid a \le r \le b, \alpha \le \theta \le \beta\}$, then $\iint_R f(x, y) \, dA = \int_\alpha^\beta \int_a^b f(r \cos\theta, r \sin\theta) r \, dr \, d\theta$. - **Theorem 2.3.2: Double Integral over General Polar Region** If $f$ is continuous on $R = \{(r, \theta) \mid r_1(\theta) \le r \le r_2(\theta), \alpha \le \theta \le \beta\}$, then $\iint_R f(x, y) \, dA = \int_\alpha^\beta \int_{r_1(\theta)}^{r_2(\theta)} f(r \cos\theta, r \sin\theta) r \, dr \, d\theta$. - **Steps for Setting up Polar Integrals:** 1. Sketch region $R$. 2. Determine $r$-limits by drawing a ray from origin, noting entry $r_1(\theta)$ and exit $r_2(\theta)$ points. 3. Determine $\theta$-limits by finding the smallest $\alpha$ and largest $\beta$ that bound $R$. #### 2.4 Applications of Double Integrals - **Volume of a Solid:** $V = \iint_R f(x, y) \, dA$ (for $f(x, y) \ge 0$). - **Area of a Plane Region:** $A = \iint_R 1 \, dA$. - **Mass of a Lamina:** For a lamina occupying region $R$ with density $f(x, y)$: $M = \iint_R f(x, y) \, dA$. - **Moments of Mass:** - About $x$-axis: $M_x = \iint_R y f(x, y) \, dA$. - About $y$-axis: $M_y = \iint_R x f(x, y) \, dA$. - **Center of Mass:** $(\bar{x}, \bar{y})$, where $\bar{x} = \frac{M_y}{M}$ and $\bar{y} = \frac{M_x}{M}$. - **Surface Area:** For a surface $S$ parametrized by $\vec{R}(u, v)$ over domain $D$: $A(S) = \iint_D \|\vec{R}_u(u, v) \times \vec{R}_v(u, v)\| \, dA$. - For a surface $z = g(x, y)$ over region $R$ in $xy$-plane: $A(S) = \iint_R \sqrt{1 + \left(\frac{\partial z}{\partial x}\right)^2 + \left(\frac{\partial z}{\partial y}\right)^2} \, dA$. #### 2.5 Triple Integrals - **Motivation: Mass Problem** Find mass of a solid $G$ with density $f(x, y, z)$. - **Definition 2.5.1: Triple Integral** For $f(x, y, z)$ over a rectangular box $G = [a, b] \times [c, d] \times [r, s]$: $\iiint_G f(x, y, z) \, dV = \lim_{\|P\| \to 0} \sum_{\ell=1}^N f(x_\ell^*, y_\ell^*, z_\ell^*)\Delta V$. - **Theorem 2.5.2: Fubini's Theorem (for Rectangular Boxes)** If $f$ is continuous on $G = [a, b] \times [c, d] \times [r, s]$, then $\iiint_G f(x, y, z) \, dV = \int_r^s \int_c^d \int_a^b f(x, y, z) \, dx \, dy \, dz$. - There are 6 possible orders of integration. - **Definition 2.5.3: Triple Integral over General Regions** For a continuous function $f$ on a solid $G$, define $F(x, y, z) = f(x, y, z)$ if $(x, y, z) \in G$ and $0$ otherwise. Then $\iiint_G f(x, y, z) \, dV = \iiint_E F(x, y, z) \, dV$ where $E$ is a box enclosing $G$. - **Types of Solids for Iterated Integrals:** 1. **Type xy:** $G = \{(x, y, z) \mid u_1(x, y) \le z \le u_2(x, y), (x, y) \in R \text{ in } xy\text{-plane}\}$. $\iiint_G f(x, y, z) \, dV = \iint_R \int_{u_1(x, y)}^{u_2(x, y)} f(x, y, z) \, dz \, dA$. 2. **Type xz:** $G = \{(x, y, z) \mid u_1(x, z) \le y \le u_2(x, z), (x, z) \in R \text{ in } xz\text{-plane}\}$. $\iiint_G f(x, y, z) \, dV = \iint_R \int_{u_1(x, z)}^{u_2(x, z)} f(x, y, z) \, dy \, dA$. 3. **Type yz:** $G = \{(x, y, z) \mid u_1(y, z) \le x \le u_2(y, z), (y, z) \in R \text{ in } yz\text{-plane}\}$. $\iiint_G f(x, y, z) \, dV = \iint_R \int_{u_1(y, z)}^{u_2(y, z)} f(x, y, z) \, dx \, dA$. - **Strategy:** Project the solid onto one of the coordinate planes to define the outer double integral, then define the inner integral based on the bounding surfaces. #### 2.6 Triple Integrals in Cylindrical Coordinates - **Cylindrical Coordinate System:** $(r, \theta, z)$. - $x = r \cos\theta$, $y = r \sin\theta$, $z = z$. - $r^2 = x^2 + y^2$, $\tan\theta = y/x$. - $r=r_0$: cylinder; $\theta=\theta_0$: plane; $z=z_0$: plane. - **Volume Element:** $dV = r \, dz \, dr \, d\theta$. - **Triple Integral in Cylindrical Coordinates:** For a solid $G$ where the projection $R$ onto the $xy$-plane is a polar region: $\iiint_G f(x, y, z) \, dV = \int_\alpha^\beta \int_{r_1(\theta)}^{r_2(\theta)} \int_{u_1(r \cos\theta, r \sin\theta)}^{u_2(r \cos\theta, r \sin\theta)} f(r \cos\theta, r \sin\theta, z) r \, dz \, dr \, d\theta$. #### 2.7 Triple Integrals in Spherical Coordinates - **Spherical Coordinate System:** $(\rho, \theta, \phi)$. - $\rho$: distance from origin ($\rho \ge 0$). - $\theta$: same as cylindrical $\theta$ (angle in $xy$-plane, $0 \le \theta \le 2\pi$). - $\phi$: angle from positive $z$-axis ($0 \le \phi \le \pi$). - $x = \rho \sin\phi \cos\theta$, $y = \rho \sin\phi \sin\theta$, $z = \rho \cos\phi$. - $\rho^2 = x^2 + y^2 + z^2$, $\tan\theta = y/x$, $\tan\phi = \frac{\sqrt{x^2+y^2}}{z}$. - $\rho=\rho_0$: sphere; $\theta=\theta_0$: half-plane; $\phi=\phi_0$: cone (or plane if $\phi_0 = \pi/2$, or axes if $\phi_0=0, \pi$). - **Volume Element:** $dV = \rho^2 \sin\phi \, d\rho \, d\phi \, d\theta$. - **Triple Integral in Spherical Coordinates:** $\iiint_G f(x, y, z) \, dV = \int_{\theta_1}^{\theta_2} \int_{\phi_1}^{\phi_2} \int_{\rho_1(\theta, \phi)}^{\rho_2(\theta, \phi)} f(\rho \sin\phi \cos\theta, \rho \sin\phi \sin\theta, \rho \cos\phi) \rho^2 \sin\phi \, d\rho \, d\phi \, d\theta$. - **Steps for Setting up Spherical Integrals:** 1. Sketch solid $G$. Draw a strip/line from origin through $G$. 2. $\rho$-limits: $r=\rho_1(\theta, \phi)$ (entry) to $r=\rho_2(\theta, \phi)$ (exit). 3. $\theta$-limits: Project $G$ onto $xy$-plane. Determine $\alpha$ to $\beta$. 4. $\phi$-limits: Angle from positive $z$-axis. Determine $\gamma$ to $\delta$. #### 2.8 Applications of Triple Integrals - **Volume of a Solid:** $V = \iiint_G 1 \, dV$. - **Mass of a Solid:** For a solid $G$ with density $f(x, y, z)$: $M = \iiint_G f(x, y, z) \, dV$. - **Moments of Mass:** - $M_{xy} = \iiint_G z f(x, y, z) \, dV$. - $M_{xz} = \iiint_G y f(x, y, z) \, dV$. - $M_{yz} = \iiint_G x f(x, y, z) \, dV$. - **Center of Mass:** $(\bar{x}, \bar{y}, \bar{z})$, where $\bar{x} = \frac{M_{yz}}{M}$, $\bar{y} = \frac{M_{xz}}{M}$, $\bar{z} = \frac{M_{xy}}{M}$. ### 3. Vector Calculus #### 3.1 Scalar and Vector Fields - **Definition 3.1.1: Scalar and Vector Fields** - A **scalar field** on $\mathbb{R}^n$ is a real-valued function whose domain is a subset of $\mathbb{R}^n$. (e.g., $f(x, y) = x^2 - 3xy + 3$). - A **vector field** on $\mathbb{R}^n$ is a function whose output is an $n$-dimensional vector, where each component is a scalar field on $\mathbb{R}^n$. - **$\mathbb{R}^2$ Vector Field:** $\vec{F}(x, y) = \langle P(x, y), Q(x, y) \rangle = P(x, y)\hat{i} + Q(x, y)\hat{j}$. - **$\mathbb{R}^3$ Vector Field:** $\vec{G}(x, y, z) = \langle P(x, y, z), Q(x, y, z), R(x, y, z) \rangle = P\hat{i} + Q\hat{j} + R\hat{k}$. - **Gradient Vector Field:** If $\phi$ is a scalar field, its gradient $\vec{\nabla}\phi$ is a vector field. - **Flowlines (Streamlines):** Curves $\vec{R}(t)$ such that their tangent vector $\vec{R}'(t)$ is equal to the vector field $\vec{F}$ at $\vec{R}(t)$. $\vec{R}'(t) = \vec{F}(\vec{R}(t))$. #### 3.2 Divergence and Curl - **Definition 3.2.1: Divergence and Curl** For $\vec{F}(x, y, z) = \langle P, Q, R \rangle$ with continuous first-order partial derivatives: 1. **Divergence of $\vec{F}$ (scalar field):** $\text{div } \vec{F} = P_x + Q_y + R_z$. 2. **Curl of $\vec{F}$ (vector field):** $\text{curl } \vec{F} = \langle R_y - Q_z, P_z - R_x, Q_x - P_y \rangle$. - **Del Operator ($\vec{\nabla}$):** $\vec{\nabla} = \left\langle \frac{\partial}{\partial x}, \frac{\partial}{\partial y}, \frac{\partial}{\partial z} \right\rangle$. - $\text{div } \vec{F} = \vec{\nabla} \cdot \vec{F}$. - $\text{curl } \vec{F} = \vec{\nabla} \times \vec{F}$ (determinant mnemonic). - **Curl for 2D Vector Field:** For $\vec{F}(x, y) = \langle P(x, y), Q(x, y) \rangle$, treat as $\vec{F}(x, y, z) = \langle P(x, y), Q(x, y), 0 \rangle$. $\text{curl } \vec{F} = \langle 0, 0, Q_x - P_y \rangle$. #### 3.3 Conservative Vector Fields - **Definition 3.3.1: Conservative Vector Field** A vector field $\vec{F}$ is conservative if it is the gradient of a scalar field $\phi$, i.e., $\vec{F} = \vec{\nabla}\phi$. $\phi$ is called a **potential function** for $\vec{F}$. - **Finding Potential Functions (2D):** If $\vec{F} = \langle P, Q \rangle$ is conservative, then $P = \phi_x$ and $Q = \phi_y$. 1. Integrate $P$ with respect to $x$ to get $\phi(x, y) = \int P \, dx + C(y)$. 2. Differentiate $\phi(x, y)$ with respect to $y$ and equate to $Q$ to find $C'(y)$. 3. Integrate $C'(y)$ with respect to $y$ to find $C(y)$. 4. Substitute $C(y)$ back into $\phi(x, y)$. (Generalizes to 3D). - **Theorem 3.3.2:** If $\phi$ and $\psi$ are potential functions for $\vec{F}$, then $\phi = \psi + K$ for some constant $K$. - **Theorem 3.3.3:** 1. If $\vec{F}$ is conservative, then $\text{curl } \vec{F} = \vec{0}$. 2. If $P, Q, R$ have continuous second-order partial derivatives, then $\text{div}(\text{curl } \vec{F}) = 0$. - **Topological Definitions:** - **Simple Curve:** Does not intersect itself (except possibly at endpoints). - **Closed Curve:** Initial and terminal points coincide. - **Connected Region:** Any two points can be joined by a continuous path within the region. - **Simply Connected Region:** A connected region where every loop can be continuously shrunk to a point within the region (no "holes"). - **Theorem 3.3.7: Test for Conservative Vector Field (3D)** Let $\vec{F}$ be a vector field on a simply-connected region with continuous first-order partial derivatives. Then $\vec{F}$ is conservative if and only if $\text{curl } \vec{F} = \vec{0}$. - **Corollary 3.3.8: Test for Conservative Vector Field (2D)** Let $\vec{F} = \langle P, Q \rangle$ be a vector field on a simply-connected region with continuous first-order partial derivatives. Then $\vec{F}$ is conservative if and only if $Q_x = P_y$. #### 3.4 Line Integrals of Scalar Fields - **Motivation: Curtain Area Problem** Find the area of a "curtain" with base curve $C$ and height $z = f(x, y)$. - **Definition 3.4.1: Line Integral with respect to Arclength ($ds$)** For $f(x, y)$ continuous on $C$ parametrized by $\vec{R}(t) = \langle x(t), y(t) \rangle$, $t \in [a, b]$: $\int_C f(x, y) \, ds = \int_a^b f(x(t), y(t)) \|\vec{R}'(t)\| \, dt = \int_a^b f(x(t), y(t)) \sqrt{(x'(t))^2 + (y'(t))^2} \, dt$. - For space curves $\vec{R}(t) = \langle x(t), y(t), z(t) \rangle$: $\int_C f(x, y, z) \, ds = \int_a^b f(x(t), y(t), z(t)) \sqrt{(x'(t))^2 + (y'(t))^2 + (z'(t))^2} \, dt$. - **Properties of Line Integrals $ds$:** - Value is independent of curve orientation: $\int_{-C} f \, ds = \int_C f \, ds$. - Value is independent of parametrization. - For piecewise smooth $C = C_1 \cup \dots \cup C_n$: $\int_C f \, ds = \sum \int_{C_i} f \, ds$. - **Applications:** Mass of a wire ($M = \int_C f \, ds$), center of mass ($\bar{x} = \frac{1}{M} \int_C xf \, ds$, etc.), arclength ($\int_C 1 \, ds$). - **Definition 3.4.2: Line Integrals with respect to $x$ and $y$** For $f(x, y)$ continuous on $C$: - $\int_C f(x, y) \, dx = \int_a^b f(x(t), y(t)) x'(t) \, dt$. - $\int_C f(x, y) \, dy = \int_a^b f(x(t), y(t)) y'(t) \, dt$. - Often combined: $\int_C P(x, y) \, dx + Q(x, y) \, dy$. - **Properties of Line Integrals $dx, dy$:** - Orientation matters: $\int_{-C} f \, dx = -\int_C f \, dx$. - For space curves, similarly: $\int_C f(x, y, z) \, dz = \int_a^b f(x(t), y(t), z(t)) z'(t) \, dt$. #### 3.5 Line Integrals of Vector Fields - **Motivation: Generalized Work Problem** Find the work done by a force field $\vec{F}$ on an object moving along a curve $C$. - **Definition 3.5.1: Line Integral of a Vector Field** For $\vec{F}$ continuous on $C$ parametrized by $\vec{R}(t)$, $t \in [a, b]$: $\int_C \vec{F} \cdot d\vec{R} = \int_a^b \vec{F}(\vec{R}(t)) \cdot \vec{R}'(t) \, dt$. - This is the work done by $\vec{F}$ along $C$. - **Relationship to Scalar Line Integrals:** - $\int_C \vec{F} \cdot d\vec{R} = \int_C P \, dx + Q \, dy + R \, dz$. - $\int_C \vec{F} \cdot d\vec{R} = \int_C (\vec{F} \cdot \vec{T}) \, ds$, where $\vec{T}$ is the unit tangent vector. #### 3.6 Fundamental Theorem of Line Integrals - **Theorem 3.6.1: Fundamental Theorem of Line Integrals (FTLI)** Let $C$ be a piecewise smooth curve from point $A$ to point $B$. Let $\phi$ be a differentiable scalar field whose gradient $\vec{\nabla}\phi$ is continuous on a region containing $C$. Then: $\int_C \vec{\nabla}\phi \cdot d\vec{R} = \phi(B) - \phi(A)$. - If $\vec{F}$ is a conservative vector field with potential function $\phi$, then $\int_C \vec{F} \cdot d\vec{R} = \phi(B) - \phi(A)$. - **Path Independence:** If $\vec{F}$ is a conservative vector field, the value of $\int_C \vec{F} \cdot d\vec{R}$ depends only on the initial and terminal points of $C$, not on the path taken. - **Theorem 3.6.2:** If $C$ is a piecewise smooth closed curve and $\vec{F}$ is a conservative vector field continuous on an open region containing $C$, then $\int_C \vec{F} \cdot d\vec{R} = 0$. - **Definition 3.6.3: Independence of Path** The line integral $\int_C \vec{F} \cdot d\vec{R}$ is independent of the path $C$ if $\int_{C_1} \vec{F} \cdot d\vec{R} = \int_{C_2} \vec{F} \cdot d\vec{R}$ for all piecewise smooth curves $C_1, C_2$ with the same initial and terminal points. - **Theorem 3.6.4: Equivalences for Path Independence** For $\vec{F}$ with continuous component functions on an open connected region $D$, the following are equivalent: (i) $\vec{F}$ is a conservative vector field on $D$. (ii) $\int_C \vec{F} \cdot d\vec{R}$ is independent of the path in $D$. (iii) $\int_C \vec{F} \cdot d\vec{R} = 0$ for every closed path $C$ in $D$. - **Methods to Evaluate $\int_C \vec{F} \cdot d\vec{R}$ when $\vec{F}$ is conservative:** 1. Find a potential function $\phi$ and use FTLI. 2. Parametrize a simpler path $C'$ from initial to terminal point and evaluate $\int_{C'} \vec{F} \cdot d\vec{R}$. #### 3.7 Green's Theorem - **Positive Orientation:** A closed curve $C$ is positively oriented if the region $R$ it encloses remains on the left as $C$ is traversed. - **Theorem 3.7.1: Green's Theorem** Let $C$ be a positively oriented piecewise-smooth simple closed curve bounding region $R$. Let $P, Q$ have continuous first-order partial derivatives on $R$. Then: $\oint_C P \, dx + Q \, dy = \iint_R \left(\frac{\partial Q}{\partial x} - \frac{\partial P}{\partial y}\right) \, dA$. - $\oint_C$ denotes a line integral around $C$ with positive orientation. - **Green's Theorem for Annular Regions:** Can be applied to regions with "holes" by appropriately orienting inner and outer boundaries. - **Corollary 3.7.2: Area using Line Integrals** The area $A$ of region $R$ enclosed by a positively oriented piecewise-smooth simple closed curve $C$ is given by: $A = \frac{1}{2} \oint_C -y \, dx + x \, dy = -\oint_C y \, dx = \oint_C x \, dy$. #### 3.8 Surface Integrals of Scalar Fields - **Motivation: Mass of a Curved Lamina** Find the mass of a curved surface $S$ with density $f(x, y, z)$. - **Definition 3.8.1: Surface Integral of a Scalar Field** For $f(x, y, z)$ continuous on a smooth surface $S$ parametrized by $\vec{R}(u, v)$ over domain $D$: $\iint_S f(x, y, z) \, d\sigma = \iint_D f(\vec{R}(u, v)) \|\vec{R}_u(u, v) \times \vec{R}_v(u, v)\| \, dA$. - $d\sigma = \|\vec{R}_u \times \vec{R}_v\| \, dA$ is the surface area element. - **For Surface $z = g(x, y)$:** Let $R$ be the projection of $S$ onto the $xy$-plane. $\iint_S f(x, y, z) \, d\sigma = \iint_R f(x, y, g(x, y)) \sqrt{1 + (g_x(x, y))^2 + (g_y(x, y))^2} \, dA$. - $\sqrt{1 + (g_x)^2 + (g_y)^2} \, dA$ is the surface area element for surfaces defined explicitly. - **Application: Mass of a Curved Lamina:** $M = \iint_S \delta(x, y, z) \, d\sigma$, where $\delta$ is the density. #### 3.9 Surface Integrals of Vector Fields - **Orientable Surface:** A surface that has a continuous unit normal vector field $\vec{n}$. - **Orientation of a Parametric Surface:** For $\vec{R}(u, v)$, the unit normal vector $\vec{n} = \frac{\vec{R}_u \times \vec{R}_v}{\|\vec{R}_u \times \vec{R}_v\|}$ defines the positive orientation. - **Orientation of a Cartesian Surface:** For $z = g(x, y)$, the upward unit normal is $\vec{n} = \frac{-g_x \hat{i} - g_y \hat{j} + \hat{k}}{\sqrt{g_x^2 + g_y^2 + 1}}$. - **Motivation: Flux Problem** Find the amount of fluid crossing a surface $S$ per unit of time, given a velocity field $\vec{F}$. - **Definition 3.9.1: Flux of a Vector Field** For $\vec{F}$ continuous on $S$ oriented by unit normal $\vec{n}$: $\iint_S \vec{F} \cdot \vec{n} \, d\sigma = \lim_{n \to \infty} \sum_{i=1}^n (\vec{F}(P_i^*) \cdot \vec{n}) \Delta\sigma_i$. - This is the surface integral of the scalar field $\vec{F} \cdot \vec{n}$. - **Calculating Flux:** - For $S$ parametrized by $\vec{R}(u, v)$ with positive orientation: $\iint_S \vec{F} \cdot \vec{n} \, d\sigma = \iint_D \vec{F}(\vec{R}(u, v)) \cdot (\vec{R}_u \times \vec{R}_v) \, dA$. - For $S: z = g(x, y)$ with upward orientation, projected onto $R$ in $xy$-plane: $\iint_S \vec{F} \cdot \vec{n} \, d\sigma = \iint_R \vec{F}(x, y, g(x, y)) \cdot \langle -g_x, -g_y, 1 \rangle \, dA$. #### 3.10 Stokes' Theorem and Gauss's Divergence Theorem - **Theorem 3.10.1: Stokes' Theorem** Let $S$ be an oriented surface whose boundary is a simple, closed curve $C$ with positive orientation. Let $\vec{F}$ be a vector field with continuous first-order partial derivatives on a region containing $S$. Then: $\oint_C \vec{F} \cdot d\vec{R} = \iint_S (\text{curl } \vec{F}) \cdot \vec{n} \, d\sigma$. - **Positive Orientation (C relative to S):** If you walk along $C$ with your head pointing in direction of $\vec{n}$, $S$ is on your left. - Stokes' Theorem generalizes Green's Theorem to 3D. - It states that the line integral around the boundary of a surface is equal to the flux of the curl of the vector field through the surface. - **Interpretation of Curl:** If $\vec{F}$ is a velocity field, $\oint_C \vec{F} \cdot d\vec{R}$ is the **circulation** of $\vec{F}$ inside $C$. The curl measures the tendency of the fluid to rotate. If $\text{curl } \vec{F} = \vec{0}$, the field is **irrotational**. - **Theorem 3.10.2: Gauss's Divergence Theorem** Let $G$ be a closed and bounded solid in $\mathbb{R}^3$ with piecewise smooth surface boundary $S$. Let $\vec{F}$ be a vector field with continuous first partial derivatives. If $\vec{n}$ represents the outward normal vector of $S$, then: $\oiint_S \vec{F} \cdot \vec{n} \, d\sigma = \iiint_G (\text{div } \vec{F}) \, dV$. - $\oiint_S$ denotes integration over a closed surface. - Gauss's Theorem states that the outward flux of a vector field through a closed surface is equal to the triple integral of the divergence of the field over the volume it encloses. - **Interpretation of Divergence:** - If $\text{div } \vec{F} > 0$ at a point, it's a **source** (fluid expands). - If $\text{div } \vec{F}