Gradient and directional derivatives

Main points:
• The gradient
○ Deltaf(x.y)=[fx(x,y),fy(x,y)]
• Properties of the gradient
○ The gradient vector always points in the direction of the greatest increase. The length of the
gradient corresponds to the steepness of the slope in that direction: the steeper the slope, the longer the gradient vector.
○ The gradient vector is always perpendicular to the level curve at which it
is rooted. All of the gradient vectors on the graph correspond to points
on the level curve f(x, y) = 1. The gradient direction is the direction
of greatest increase. The tangent to the level curve is the direction of no change.
○ The opposite direction to that of the gradient is the direction of greatest
decrease.
• Directional derivatives
○ Allows to find the rate of change in any direction in the (x,y) -plane
○ Directional derivative of f(x,y) at the point (x0,y0) in the direction of a unit vector u=(u1,u2) is
§ Duf(x0,y0)=fx(x0,y0)u1+fy(x0,y0)u2

Challenges:
I understood the method how to find gradients and directional derivatives. I am still lacking the full understanding of the importance of gradient and directional derivatives in real world problems. I also had problems understanding the properties of the gradient. I am hoping that going through the topic tomorrow in class will help me to understand these areas more.

Reflection:
The lack of real world examples continued and I hope to come across more practical examples with the after class problems. I think the current topic is very relevant in the area of economics, but I am hoping still to see the related examples.