Welcome to

APMA E2000

Multivariable Calculus for Engineers and Applied Scientists

Drew Youngren dcy2@columbia.edu

Welcome to

APMA E2000

MVC

Drew Youngren dcy2@columbia.edu

Welcome to

APMA E2000

Multi

Drew Youngren dcy2@columbia.edu

Logistics

Courseworks

aka Canvas

courseworks2.columbia.edu

  • Syllabus
  • Assignments
  • Links
  • Grades

Gradescope

gradescope.com

  • Submit HW
  • Take quizzes

Ed Discussions

edstem.org

  • Announcements
  • Discussion
  • Q&A

JupyterLite (Experimental)

  • Computational environment
  • Extra resources
3Demos Logo

3demos.ctl.columbia.edu

  • Visualization
  • Interaction
  • Bells & Whistles

Spaces

$\mathbb{R}$    The Real Numbers

aka "scalars". The complete, ordered field.

$\mathbb{R}^2$    The Cartesian Plane

\[ \mathbb{R}^2 = \{ (x,y) \mid x,y \in \mathbb{R} \} \]

$\mathbb{R}^3$    3-space

\[ \mathbb{R}^3 = \{ (x,y,z) \mid x,y,z \in \mathbb{R} \} \]

$\mathbb{R}^n$    $n$-space

\[ \mathbb{R}^n = \{ (x_1, \ldots, x_n) \mid x_i \in \mathbb{R} \} \]
Definition

Let $P = (p_1, \ldots, p_n)$ and $Q = (q_1, \ldots, q_n)$ be two points in $\mathbb{R}^n$. The distance formula is:

\[ d(P, Q) = \sqrt{\sum_{i = 1}^n (p_i - q_i)^2} \]

Vectors

Definition

A vector is a directed line segment in \(\mathbb{R}^n\) modulo location.

Two vectors are considered the same if they have the same magnitude and direction.

Vector Notation

\[ \mathbf v = \vec v = \langle v_1, v_2, \ldots, v_n \rangle = \begin{bmatrix} v_1 \\ v_2 \\ \vdots \\ v_n \end{bmatrix} \]

The $v_i$'s are the components of $\mathbf{v}$, the displacements in each dimension.

Vector Operations

Magnitude

\[|\mathbf{v}| = \sqrt {\sum_{i=1}^n v_i^2}\]

The magnitude or length of a vector is a nonnegative scalar.

It is non-degenerate which just means \[ \mathbf v = \mathbf 0 \iff |\mathbf v| = 0.\]

Scaling

Scalar multiplication, or really scalar-vector multiplication, simply scales each component by a common factor.

\[ c\mathbf v = \left\langle c v_1, c v_2, \ldots, c v_n \right\rangle \]

Question

What happens to the magnitude when of a vector when scaling?

$ |c \mathbf{v} | = $ $ |c \mathbf{v} | = c|\mathbf v| $ $ |c \mathbf{v} | = \sout{c|\mathbf v|} $ $ |c \mathbf{v} | = |c||\mathbf v| $
Application

Normalization

A unit vector has magnitude 1. \[|\mathbf u| = 1\]

Every nonzero vector $\mathbf v \neq \mathbf 0$ can be normalized by dividing by its magnitude.

\[ \mathbf u = \frac{\mathbf v}{|\mathbf v|} \]
\[ \mathbf u = \frac{\mathbf v}{|\mathbf v|} = \frac{1}{|\mathbf v|} \mathbf v \]

Addition

Vector addition works exactly how one thinks it would.

\[ \mathbf v + \mathbf w = \left\langle v_1 + w_1, \ldots, v_n + w_n \right\rangle \]
Geometrically, we refer to this as tip-to-tail addition.

A sum of scaled vectors is called a linear combination and is the fundamental vector expression.

\[ a \mathbf v + b \mathbf w \] \[ \sum_{i = 1}^m c_i\mathbf v_i\]

Vector Uses

Points or Vectors?

\[(22, 3) \leftrightarrow \left\langle 22, 3 \right\rangle\]
22nd St & 3rd Ave $\leftrightarrow$ 22 blocks N, 3 blocks W

A position vector (or vectors) can be used to specify a point (or set of points).

Question

What location is halfway between 22nd St & 3rd Ave and 42nd & 5th?

Question

What location is halfway between 22nd St & 3rd Ave and 42nd & 5th?

\[ \mathbf p = \left\langle 22, 3 \right\rangle \qquad \mathbf q = \left\langle 42, 5 \right\rangle\ \]
\[ \mathbf m = \mathbf p + \frac12 (\mathbf q - \mathbf p) \]
\[ = \frac{\mathbf p + \mathbf q}{2} = \left\langle 32, 4 \right\rangle\]
Question

What location is $\frac34$ of the way from 22nd St & 3rd Ave to 42nd & 5th?

\[ \mathbf x = \mathbf p + \frac34 (\mathbf q - \mathbf p) \]
\[ = \frac34\mathbf q + \frac14\mathbf p = \text{37th \& Madison} \]
Definition
Special linear combinations of the form \[ (1 - t) \mathbf v + t \mathbf w \] for some $0 \leq t \leq 1$ are called convex combinations.

3Demos example

Question

What is this linear combination?

\[ \frac14 \mathbf h + \frac14 \mathbf q + \frac{3}{10} \mathbf m + \frac15 \mathbf f \]

See syllabus.

Example

Write a vector equation for the position on the sphere centered at $\mathbf p = \left\langle p_1, p_2, p_3 \right\rangle$ with radius $R$.

Solution: \[|\mathbf x - \mathbf p | = R \]

Note: This is exactly the distance formula. We use the vector variable $\mathbf x$ here to represent all the positions. We will do that a lot.

Example

Write an equation for the set of points whose distances to the origin and to $(2, 0, 0)$ sum to $4$.

Solution: \[|\mathbf x | + | \mathbf x - \left\langle 2, 0, 0 \right\rangle | = 4 \]

But what is this set? 3Demos

Learning Outcomes

You should be able to...
  • Link relations on variables to sets in Euclidean space (coordinate axes, planes, etc.).
  • Evaluate the results of vector operations formulaically and visually.
  • Use linear combinations to specify positions in relation to one another.
  • Express linear interpolations and weighted averages as convex combinations.