Notes on Vectors p 4-8

Pages 4 - 8, more on vectors!!!

1) Main Points

The first section of Page 1 reiterates what we learned in class about the dot product. A "Gradient" is a vector that consists of the partial derivatives f(x,y). It is written as delta f(x,y) or grad(f). To find a gradient you find the partial derivatives of the equation and put them next to each-other separated by a comma, for example delta f(x,y)= <3x, 4x^2>. Then to find the gradient at a certain point you substitute in the desired x and y values to get a vector such as <4, 12>. Next the reading introduces "Directional Derivatives" which go beyond partial derivates and calculates how the function changes when moved in any direction instead of either x or y. The Directional derivative at a given point in given direction is written as Df(x0,y0) = the gradient of f at (x0, y0) times (dot product) the vector u. The reading goes on to justify this algebraically. It shoes a contour diagram of a multivariable function is shown of along with several gradient vectors at different points. The gradient vector always points in the direction of greatest increase, and its length corresponds with how steep the slope is.It helps to think of this as a mountain, with the gradient vector pointing "Up hill".


2) Challenges

I'm Still a bit confused about unit vectors, the reading made some connections about unit vectors that I didn't understand. I am unclear on what the directional derivatives actually tell us. To find gradients and Directional Derivatives it is important to have a really good hold on derivatives in general because messing them up will affect what comes later.

3) Reflections

This chapter helps explain what vectors are useful for, they can visually explain a lot in a graph or a contour diagram, especially about rates of change and derivatives. Seeing them on a contour diagram really helped me see how they work and what they can show.