## Calculus 3 (the Best Way to Get Started)

Does your interest in math extend beyond bland proofs and endless theory? Well, say goodbye to convoluted explanations and hello to clear examples and practice exercises. In Multivariable Calculus (Calculus 3), you’ll find a teaching method meant to give advanced

## Vector Functions and Space Curves

Vector functions can be used to model the motion of an object. You may want the object’s path (geometry only), but you may also want to know its speed, direction of motion, and acceleration at each point in time, in

## Derivatives and Integrals of Vector Functions

We study differentiation and integration of vector functions of a single variable. Tangent vectors and unit normal vectors are also considered. Several exercises are given at the end. Derivatives of Vector Functions As with functions of one variable we define

## Velocity and Acceleration (of an Object)

We discuss the motion of an object in three dimensions. We study the path of an object, and its positions at various times, its velocity, speed and direction of travel of the object along the path. The acceleration at any

## Arc Length and Curvature of Smooth Curves

We discuss piecewise smooth vector functions of one variable. In particular we study arc length and curvature. We define the curvature of a vector-function in term so the unit tangent vector. We give several concrete examples of how to determine

## Multivariable Functions (and Their Level Curves)

A real-valued function of several real variables is a function that takes as input real numbers (represented by variables) and produces another real number, the value of the function. We discuss the basic of multivariable functions such as find the

## Continuous Function and Multivariable Limit

Intuitive Meaning of Multivariable Limit Recall when considering $$\lim_{x\to c} f(x)=L$$ we need to examine the approach of $x$ to $c$ from two directions –namely the left-hand limit and the right-hand limit. However for functions of two variables, we write

## Partial Derivatives (and Partial Differential Equations)

Partial Derivatives For the partial differentiation of a function of two variables, $z=f(x,y)$, we find the partial derivative with respect to $x$ by regarding $y$ as a constant while differentiating the function with respect to $x.$ Similarly, the partial derivative

## Differentials and the Total Differential

Differentials Definition. Let $z=f(x,y)$ and let $\Delta x$ and $\Delta y$ be increments of $x$ and $y$, respectively. The differentials $dx$ and $dy$ of the independent variables $x$ and $y$ are defined by $dx=\Delta x$ and $dy=\Delta y.$ The differential

## Chain Rule for Multivariable Functions

Introduction to the Chain Rule Recall that the chain rule for functions of a single variable gives the rule for differentiating a composite function: if $y=f(x)$ and $x=g(t),$ where $f$ and $g$ are differentiable functions, then $y$ is a a

## Directional Derivatives and Gradient Vectors

Definition of Directional Derivative Partial derivatives find the rate of change of $z=f(x,y)$ in the directions of the $x$ and $y$ axis; that is in the direction of the unit vectors ${i}$ and ${j}$, respectively. Definition. Let $f$ be a

## Normal Lines and Tangent Planes

Tangent Planes Suppose $S$ is a surface with the equation $z=f(x,y)$ where $f$ has continuous first partial derivatives $f_x$ and $f_y.$ Let $P(a,b,c)$ be a point on $S$ and let $C_1$be the curve of intersection of $S$ with the plane

## Relative Extrema and the Second Partials Test

Relative Extrema Definition. Let $f$ be a function of two variables $x$ and $y.$ The function $f$ has a relative maximum at $\left(x_0,y_0\right)$ if $f(x,y)\leq f\left(x_0,y_0\right)$ for all $(x,y)$ in an open disk containing $\left(x_0, y_0\right).$ The function $f$ has

## Absolute Extrema (and the Extreme Value Theorem)

Absolute Maximum and Absolute Minimum Definition. Let $f$ be a function of two variables $x$ and $y.$ (1) The function $f$ has an absolute maximum at $\left(x_0,y_0\right)$ if $f(x,y)\leq f\left(x_0,y_0\right)$ for all $(x,y)$ in the domain $D$ of $f.$ (2)

## Lagrange Multipliers (Optimizing a Function)

In many applied problems, the main focus is on optimizing a function subject to constraint; for example, finding extreme values of a function of several variables where the domain is restricted to a level curve (or surface) of another function

## Double Integrals and the Volume Under a Surface

The Volume Under a Surface Consider the rectangle given by $$R=[a,b]\times[c,d]=\{(x,y)\mid a\leq x\leq b, c\leq x\leq d\}.$$ We wish to construct a (regular) partition of $R.$ To do so, let  a=x_0 < x_1 < x_2 < \cdots <

## Jacobian (Change of Variables in Multiple Integrals)

Jacobians If $x=x(u,v)$ and $y=y(u,v)$ then the Jacobian of $x$ and $y$ with respect to $u$ and $v$ is \begin{align*} \frac{\partial (x,y)}{\partial (u,v)} & =J(u,v) \\ & = \left| \begin{array}{cc} \frac{\partial x}{\partial u} & \frac{\partial x}{\partial v} \\ \frac{\partial y}{\partial

## Vector Fields and Gradient Fields

Introduction to Vector Fields Definition. A vector field in $\mathbb{R}^n$ is a function $\mathbf{V}$ that assigns a vector from each point in its domain. A vector field with domain $D$ in $\mathbb{R}^n$ has the form {V}(x_1, \ldots, x_n) =

## Divergence and Curl of a Vector Field

The Divergence and Curl Definition. Let $\mathbf{V}$ be a given vector field. The divergence of $\mathbf{V}$ is defined by div $\mathbf{V}=\nabla \cdot \mathbf{V}$ and the curl of $\mathbf{V}$ is defined by curl $\mathbf{V}=\nabla \times \mathbf{V}$ where \nabla =\frac{\partial }{\partial