Seminar 9

So this day, we have been working on some very interesting topics. It was actually the first day that I attended a Math-seminar and it was long overdue.

The first exercise was about contour lines. So, let us step one step back. Traditionally, we have learnt to see to coordinate system in 2D-dimensions. But originally the coordinate system was meant to be viewed in a 3D-dimensions.
Here it was supposed that we had 3 main coordination lines: x, y, z where x would be the length, y would diagnose and z would the height. Assuming this and in order to view the coordinate system in a 2D-dimension, we have to assume that we look on coordination-system from above.

So what is a contour line?

I would say that a contour line is a line of body illustrated in a 3-dimensions room. If we translate it into a 2-dimensional room, the contour line is actually what would describe a graph or a function of two variables which would be x,y as we look on the system from above (thus "ignoring" z)

Better: a contour line is the reduced illustration of where the xy-plane touches/crosses the body in 3D-dimensions.

So if I have a function of f(x,y)=sin(y)+y+ex

a. P(0,0) belongs to which contour line. So actually we could see the contour line as a result of where xy touch the body. So if x and y are of a certain value then contour line would be the output of these values. Consequently, the contour line when x=0 y=o would be 1.

b. Justify that this contour line admits an equation of the form y=..(x) around P. What is the slope of ..'(0) of this contour line at x=0