Observation 1-Syllabus

Remember

Brussel has a HW that they upload by Monday.
Doesn't ornament canvas, just the link to the dropbox of all the files.
Does a HW checkup on Friday, where he quizzes on just one randomly selected problem form the course.
Will choose people at random.

Lecture Main Points

Defines what "Multivariable Calculus" means
Tries to give some overarching summary of what they'll cover in the course

Course Goal

Define a surface $S : z = f (x, y)$ in the xyz-space
We can have a vector field $\vec{F} (x, y)$ that goes through the surface, to find the flux of the vector field through $S$
They should read 12.6 and 14.1 by Tuesday...

Functions

Given a function $f (x, y)$ gives a single output $z$ in 3D space. Similarly, we can have $g (x, y, z)$ .
Some examples include:

f (x, y) = x^{2} + y^{2}, g (x, y) = \frac{1}{\sqrt{x^{2} + y^{2}}}, . . .

The analysis will change from calculus in single variables to multiple variables.

Defines:

Domain
- ignore $\sqrt{}$ of negatives
- having $0$ in the denominator
- having nonnegative $\log$ input arguments
Codomain
Range

Example

Find the domain and range of:

f (x, y) = \frac{1}{\sqrt{x^{2} - y^{2}}}

Note that we require that:

\sqrt{x^{2} - y^{2}} \geq 0, \sqrt{x^{2} - y^{2}} \neq 0 \Rightarrow \sqrt{x^{2} - y^{2}} > 0

Thus:

\Rightarrow x^{2} - y^{2} > 0

x^{2} > y^{2}

| x | > | y |

They should:

Graph out the equality
Try trial points to shade

The range is $R = R^{+}$

What is the graph for this function? To draw this:
- Set a 'height' value for $z$
- Solve for $y$ in the $f (x, y) = z$ equation
- Plot the $y$ graph, and repeat for various $z$ constant values

Note that 12.6 (what they have to read) contains a lot of the hyperbola/ellipse/parabola definitions that they have to know. They'll define more formally contour diagrams tomorrow's lecture.