Calculus III Formulas

Cheatsheet Content

### Vectors and the Geometry of Space - **Vector:** $\vec{v} = \langle v_1, v_2, v_3 \rangle$ or $\vec{v} = v_1\mathbf{i} + v_2\mathbf{j} + v_3\mathbf{k}$ - **Magnitude of a Vector:** $|\vec{v}| = \sqrt{v_1^2 + v_2^2 + v_3^2}$ - **Unit Vector:** $\hat{u} = \frac{\vec{v}}{|\vec{v}|}$ - **Distance Formula in 3D:** $D = \sqrt{(x_2-x_1)^2 + (y_2-y_1)^2 + (z_2-z_1)^2}$ - **Equation of a Sphere:** $(x-h)^2 + (y-k)^2 + (z-l)^2 = r^2$ ### The Dot Product - **Definition:** $\vec{a} \cdot \vec{b} = a_1b_1 + a_2b_2 + a_3b_3$ - **Geometric Definition:** $\vec{a} \cdot \vec{b} = |\vec{a}||\vec{b}|\cos\theta$ - **Angle between Vectors:** $\cos\theta = \frac{\vec{a} \cdot \vec{b}}{|\vec{a}||\vec{b}|}$ - **Orthogonal Vectors:** $\vec{a} \cdot \vec{b} = 0 \iff \vec{a} \perp \vec{b}$ - **Scalar Projection of $\vec{b}$ onto $\vec{a}$:** $comp_{\vec{a}}\vec{b} = \frac{\vec{a} \cdot \vec{b}}{|\vec{a}|}$ - **Vector Projection of $\vec{b}$ onto $\vec{a}$:** $proj_{\vec{a}}\vec{b} = \left(\frac{\vec{a} \cdot \vec{b}}{|\vec{a}|^2}\right)\vec{a}$ ### The Cross Product - **Definition:** $\vec{a} \times \vec{b} = \begin{vmatrix} \mathbf{i} & \mathbf{j} & \mathbf{k} \\ a_1 & a_2 & a_3 \\ b_1 & b_2 & b_3 \end{vmatrix}$ - **Magnitude:** $|\vec{a} \times \vec{b}| = |\vec{a}||\vec{b}|\sin\theta$ - **Area of Parallelogram:** $A = |\vec{a} \times \vec{b}|$ - **Volume of Parallelepiped:** $V = |\vec{a} \cdot (\vec{b} \times \vec{c})|$ (Scalar Triple Product) ### Equations of Lines and Planes #### Lines - **Vector Equation:** $\vec{r}(t) = \vec{r}_0 + t\vec{v}$ - **Parametric Equations:** $x = x_0 + at$, $y = y_0 + bt$, $z = z_0 + ct$ - **Symmetric Equations:** $\frac{x-x_0}{a} = \frac{y-y_0}{b} = \frac{z-z_0}{c}$ #### Planes - **Vector Equation:** $\vec{n} \cdot (\vec{r} - \vec{r}_0) = 0$ - **Scalar Equation:** $a(x-x_0) + b(y-y_0) + c(z-z_0) = 0$ - **General Equation:** $ax + by + cz + d = 0$ - **Distance from Point to Plane:** $D = \frac{|ax_0 + by_0 + cz_0 + d|}{\sqrt{a^2+b^2+c^2}}$ ### Cylinders and Quadric Surfaces - **General Equation:** $Ax^2+By^2+Cz^2+Dxy+Eyz+Fxz+Gx+Hy+Iz+J=0$ - Examples: Ellipsoid, Hyperboloid (of one or two sheets), Cone, Elliptic Paraboloid, Hyperbolic Paraboloid ### Vector Functions and Space Curves - **Vector Function:** $\vec{r}(t) = \langle f(t), g(t), h(t) \rangle$ - **Derivative:** $\vec{r}'(t) = \langle f'(t), g'(t), h'(t) \rangle$ - **Tangent Vector:** $\vec{r}'(t)$ - **Unit Tangent Vector:** $\vec{T}(t) = \frac{\vec{r}'(t)}{|\vec{r}'(t)|}$ ### Arc Length and Curvature - **Arc Length:** $L = \int_a^b |\vec{r}'(t)| dt$ - **Curvature:** $\kappa = \frac{|\vec{T}'(t)|}{|\vec{r}'(t)|} = \frac{|\vec{r}'(t) \times \vec{r}''(t)|}{|\vec{r}'(t)|^3}$ - **Principal Unit Normal Vector:** $\vec{N}(t) = \frac{\vec{T}'(t)}{|\vec{T}'(t)|}$ - **Binormal Vector:** $\vec{B}(t) = \vec{T}(t) \times \vec{N}(t)$ ### Motion in Space: Velocity and Acceleration - **Position Vector:** $\vec{r}(t)$ - **Velocity Vector:** $\vec{v}(t) = \vec{r}'(t)$ - **Speed:** $|\vec{v}(t)| = |\vec{r}'(t)|$ - **Acceleration Vector:** $\vec{a}(t) = \vec{v}'(t) = \vec{r}''(t)$ - **Tangential Component of Acceleration:** $a_T = \frac{\vec{v} \cdot \vec{a}}{|\vec{v}|} = \frac{d}{dt}|\vec{v}|$ - **Normal Component of Acceleration:** $a_N = \frac{|\vec{v} \times \vec{a}|}{|\vec{v}|} = \kappa |\vec{v}|^2$ - **Total Acceleration:** $\vec{a} = a_T \vec{T} + a_N \vec{N}$ ### Functions of Several Variables - **Function:** $z = f(x,y)$ or $w = f(x,y,z)$ - **Level Curves:** $f(x,y) = k$ (constant) - **Level Surfaces:** $f(x,y,z) = k$ (constant) ### Partial Derivatives - **Definition:** - $f_x(x,y) = \lim_{h \to 0} \frac{f(x+h,y) - f(x,y)}{h}$ - $f_y(x,y) = \lim_{h \to 0} \frac{f(x,y+h) - f(x,y)}{h}$ - **Notation:** $f_x = \frac{\partial f}{\partial x}$, $f_y = \frac{\partial f}{\partial y}$ - **Higher-Order Partials:** $f_{xx}, f_{xy}, f_{yx}, f_{yy}$ - **Clairaut's Theorem (Equality of Mixed Partials):** If $f_{xy}$ and $f_{yx}$ are continuous, then $f_{xy} = f_{yx}$. ### Tangent Planes and Linear Approximations - **Equation of Tangent Plane to $z=f(x,y)$ at $(x_0, y_0, z_0)$:** $z - z_0 = f_x(x_0, y_0)(x - x_0) + f_y(x_0, y_0)(y - y_0)$ - **Linear Approximation (Linearization):** $L(x,y) = f(x_0, y_0) + f_x(x_0, y_0)(x - x_0) + f_y(x_0, y_0)(y - y_0)$ $f(x,y) \approx L(x,y)$ - **Differential:** $dz = f_x(x,y)dx + f_y(x,y)dy$ ### The Chain Rule - **Case 1 (z = f(x,y), x = g(t), y = h(t)):** $\frac{dz}{dt} = \frac{\partial z}{\partial x}\frac{dx}{dt} + \frac{\partial z}{\partial y}\frac{dy}{dt}$ - **Case 2 (z = f(x,y), x = g(s,t), y = h(s,t)):** $\frac{\partial z}{\partial s} = \frac{\partial z}{\partial x}\frac{\partial x}{\partial s} + \frac{\partial z}{\partial y}\frac{\partial y}{\partial s}$ $\frac{\partial z}{\partial t} = \frac{\partial z}{\partial x}\frac{\partial x}{\partial t} + \frac{\partial z}{\partial y}\frac{\partial y}{\partial t}$ - **Implicit Differentiation:** If $F(x,y) = 0$, then $\frac{dy}{dx} = -\frac{F_x}{F_y}$ - **Implicit Differentiation (3 variables):** If $F(x,y,z) = 0$, then $\frac{\partial z}{\partial x} = -\frac{F_x}{F_z}$ and $\frac{\partial z}{\partial y} = -\frac{F_y}{F_z}$ ### Directional Derivatives and the Gradient Vector - **Gradient Vector:** $\nabla f(x,y) = \langle f_x(x,y), f_y(x,y) \rangle$ or $\nabla f(x,y,z) = \langle f_x, f_y, f_z \rangle$ - **Directional Derivative:** $D_{\vec{u}}f(x,y) = \nabla f(x,y) \cdot \vec{u}$, where $\vec{u}$ is a unit vector. - **Maximum Rate of Increase:** $|\nabla f|$ in the direction of $\nabla f$. - **Minimum Rate of Decrease:** $-|\nabla f|$ in the direction of $-\nabla f$. - **Gradient is Normal to Level Curves/Surfaces:** $\nabla f$ is orthogonal to the level curve $f(x,y)=k$ or level surface $f(x,y,z)=k$. ### Maximum and Minimum Values - **Critical Points:** Occur where $\nabla f = \vec{0}$ or where $f_x$ or $f_y$ do not exist. - **Second Derivatives Test:** For a critical point $(a,b)$ where $f_x(a,b)=0$ and $f_y(a,b)=0$, let $D = D(a,b) = f_{xx}(a,b)f_{yy}(a,b) - [f_{xy}(a,b)]^2$. - If $D > 0$ and $f_{xx}(a,b) > 0$, then $f(a,b)$ is a local minimum. - If $D > 0$ and $f_{xx}(a,b) ### Lagrange Multipliers - **To find max/min of $f(x,y,z)$ subject to constraint $g(x,y,z)=k$:** Solve $\nabla f = \lambda \nabla g$ and $g(x,y,z)=k$ for $x, y, z, \lambda$. - **System of Equations:** $f_x = \lambda g_x$ $f_y = \lambda g_y$ $f_z = \lambda g_z$ $g(x,y,z) = k$