14.6: Directional Derivatives and the Gradient Vector (1/2)

Objectives:
11. Define the directional derivative.
12. Define and compute the gradient of a function.
13. Know that the gradient at a point is orthogonal to the level set containing the point.
14. Find the direction in which a function increases most rapidly and find the directional derivative in that direction.

[@sgt.sargent3242]

This semester I learned that I'm a visual learner because I love watching youtube videos and taking supplementary lecture notes. These lecture series vids have been a life saver for me in calc 3. I officially have 3 notebooks now, one for class lectures, one for youtube lectures, and one for practice problems. That plus reviewing the class slides have saved me.

[@user-ke4ip6dl8v]

calc 3 has been brutal and you lecture on the same textbook as my class. you videos are super helpful so please never stop making them! thank you so much!!!!!

[@sidrafatima7590]

Ca you tell me what textbook she is using in this video ?

[@petraescajeda5403]

@@sidrafatima7590 seems to be calculus early transcendentals by james stewart

[@anekleynhans1991]

@@sidrafatima7590 Calculus metric version by James Stewart, Daniel Clegg and Saleem Watson

[@jameyatesmauriat6116]

I can’t imagine you’re holding the pen like this and could write anything!

[@dunyamiletic788]

I am so thankful that I found your channel! You are soooo much better than all of my university professors

[@ChefFarisMom3]

I can’t thank you for this mercy on us. Loads of respect, and prayers.

[@ellie-anne]

Thank you sop much for your videos! I am studying at Vanderbilt to be a high school math teacher and I would've failed out of Calc 3 if it wasn't for you! I wish I could be in your classes! You present the material on such a digestible manner and I love how you constantly check in with your students to see if they are understanding it. You are an example to me of how I want to teach in the future! PS: I am also from the suburbs of Chicago! You remind me of my Wheaton North High School math teachers - they were also amazing!

[@alexandraniedden5337]

This was such a sweet comment. Thank you for taking the time to write to me. Good luck in your studies!

[@edith4real]

Hey, I want to go to Vanderbilt, and I am currently taking calculus 3. Are the math classes difficult there?

[@reconvene4]

Hi! Your videos have been extremely helpful. Do you have any plans to create a Patreon or any other way we can support you? When spending thousands on tuition, helping professors who do a genuinely amazing job explaining topics seems like a no brainer.

[@alexandraniedden5337]

Great, I'm glad to hear some of the videos have been helpful! I have no plans to create a Patreon account. These videos are nothing fancy, just recordings of my in-class lectures. I am happy to share them though, as I certainly remember sitting through some unclear college lectures. Good luck!

[@balvsmalvs5425]

Very happy to have found out about your videos. Happy to listen to your lectures. You're a good teacher.

[@CHAYITO-ii5pt]

THANK YOU FOR SHARING YOUR KNOWLEDGE, I APPRECIATE IT. LOVE HOW YOU LECTURE, YOU MAKE IT SIMPLE TO COMPREHEND 🙏🏼🤓

[@mahbubakhanam9659]

At 38:01 where did the -1 come from? I'm a little confused right there.

[@SMSHB]

You are an awesome teacher, I like your way of teaching. Could you please provide the name of the text book that you shown in the video.

[@es8336]

hello what is the answer of the question at 24:52

[@khushbakhtawan9467]

why is it (1+x^2+2y^2+3y^2)^2 what happened to the squareroot for the last example?

[@ZinzinsIA]

Absolutely amazing video ,thank you so much it is really well done. Just a thing to be sure : when we say the gradient is by itself a direction and the moreover the direction of steepest ascent (and not descent), behind there is the idea that the vector is calculated in relation with the vectors of the basis right ? It is linked to the basis orientation ? I mean I try to understand why gradient which is just a slope as you say at the beginning because partial derivatives can go in both direction, why it naturally points towards the steepest ascent and not descent. I think that with another orientation, let's say e1 = (-1, 0, 0) for example, it would naturally points towards the steepest descent right ? Besides, in the traditional basis, -gradient points toward the steepest descent

[@spdas5942]

Madam, what's the name and author of the text you are using in your lecture, please ? You are really a great teacher . God bless you mam .

[@alexandraniedden5337]

I use Stewart's Multivariable Calculus (8th edition).

[@markmaraia7513]

Thank you for all of your help

[@nailahafeez2333]

Is it from Thomas calculus book?

[@alexandraniedden5337]

No, I teach from Stewart's Multivariable Calculus.

[@wieer6176]

So i have questions if creases and increasec

[@kavumapaul-sy8nr]

Is gradient the same as direction of increase or decrease?

[@alexandraniedden5337]

This is one use of the gradient, yes. The gradient gives us the direction of maximum increase (not decrease).

[@janoycresva276]

Today is exactly 1 year from when this video was published.

[@jvzmeen]

and today is exactly 4 yrs

[@mohd55531]

can you tell me the name of the book and the author of it please ?

[@alexandraniedden5337]

I use Zill's 9th edition Multivariable Calculus.

[@mohd55531]

@@alexandraniedden5337 thanks

[@Festus2022]

"h" doesn't go to infinity. It goes to ZERO

[@ishamenon2836]

in the last problem, where does the 5/8 come from?

[@alexandraniedden5337]

In the line before the 5/8, I factored out the value 160/(1+x^2+2y^2+3y^2)^2. If we substitute the point (1, 1, -2) into that factor, we get 5/8.

[@ishamenon2836]

@@alexandraniedden5337 This makes sense, thank you!

[@calebereid]

Thanks for the help so much but how is 0 times cos(0) equal to the 3? It is at 31:45.

[@alexandraniedden5337]

Thank you for providing the timestamp. That 3 comes from xycos(yz) (the 3rd component of the gradient from part a). Using the point (1, 3, 0), we then have 1*3cos(3*0) =3. I hope that helps!

[@calebereid]

@@alexandraniedden5337 Oh okay thanks.

[@geraldcalero4179]

How did you get the unit vector at 24:02 ?

[@alexandraniedden5337]

I found the unit vector using the angle (theta) given. The unit vector can be found using (think: unit circle).

[@geraldcalero4179]

@@alexandraniedden5337 thanks I figured it out this way as well appreciate it a lot !