15.5 Directional Derivatives MAT 3238 Vector Calculus 15.5 Directional Derivatives

Homework Both written and WA HW due ???day. I suggest you do the written part first. Let me remind you that your only reference are The textbook The lecture note Do not google or look up other references

Steepest Descent Algorithm Minimize Time – path planning for a robot in Mars Minimize Energy – choose parameters so that the energy is min.: Most stable All kind of applications!

Steepest Descent Algorithm At each time step, need to choose a direction where the changes is the greatest. Changes is measure by slope (i.e. derivative) of the tangent along that direction. We need a notion of slopes along different directions.

Derivatives Along Directions We know how to do this in the 𝑥 and 𝑦 directions:

Derivatives Along Directions How to extend this notion to other directions?

Directional Derivatives The directional derivative of 𝑓 at a point in the direction of a unit vector 𝑢= 𝑎,𝑏 is if this limit exists.

Directional Derivatives In particular, if... 𝑢= 1,0 , 𝐷 𝑢 𝑓 𝑥 0 , 𝑦 0 = 𝑢= 0,1 , 𝐷 𝑢 𝑓 𝑥 0 , 𝑦 0 = So, this definition is in consistent with what we know.

FAQ Q: We hate limits. Can we use a formula to compute the directional derivatives? A: Yes, of course!

Directional Derivatives

Observation #1 𝐷 𝑢 𝑓 is a linear combinations of 𝑓 𝑥 and 𝑓 𝑦 .

Observation #2 𝐷 𝑢 𝑓 is a linear combinations of 𝑓 𝑥 and 𝑓 𝑦 . 𝑎, 𝑏 behave “almost” like “proportions”

Example 1

Why are They Believable? Fake Formulas?

The Proof

The Proof