How do you sketch level curves of multivariable functions?

► My Partial Derivatives course: www.kristaking...
In this video we're talking about how to sketch the level curves of a multivariable function.
Skip to section:
0:48 // What are level curves?
4:03 // How to sketch level curves?
5:02 // How to find equations of level curves?
7:52 // How to rewrite the function to graph the multivariable function?
12:15 // How many level curves should you draw?
13:30 // What are the level curves for a plane?
16:24 // Summary for how to sketch level curves
Whenever you're dealing with a multivariable function, the graph of that function will be a three-dimensional figure in space. If you take a perfectly horizontal sheet or plane that's parallel to the xy-plane, and you use that to slice through your three-dimensional figure, then what you get at the intersection of the figure and the plane is a two-dimensional curve.
What we want to be able to do is slice through the figure at all different heights in order to get what we call the "level curves" of a function. Then we want to be able to transfer all those two-dimensional curves into the two-dimensional plane, sketching those in the xy-plane.
This will give us the sketch of level curves of the function. In this video we're going to talk about how to find the level curves both graphically (by looking at a picture of the three-dimensional figure) and algebraically, by replacing z in the multivariable function with a constant c, and then substituting different values for c in order to get equations that are in terms of x and y only and can therefore be graphed in the xy-plane.
Music by Joakim Karud: / joakimkarud
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Hi, I’m Krista! I make math courses to keep you from banging your head against the wall. ;)
Math class was always so frustrating for me. I’d go to a class, spend hours on homework, and three days later have an “Ah-ha!” moment about how the problems worked that could have slashed my homework time in half. I’d think, “WHY didn’t my teacher just tell me this in the first place?!”
So I started tutoring to keep other people out of the same aggravating, time-sucking cycle. Since then, I’ve recorded tons of videos and written out cheat-sheet style notes and formula sheets to help every math student-from basic middle school classes to advanced college calculus-figure out what’s going on, understand the important concepts, and pass their classes, once and for all. Interested in getting help? Learn more here: www.kristakingm...
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[@Bm23CC]

best explanation of level curves ever.

[@NiggaSlayer666]

you right, calculator.

[@mohammedjawahri5726]

I can't believe i watched like 6 videos on this and NOBODY thought of putting the first diagram she put to the left (3 dimensional graph combining both the function and level curve), i've been trying to understand this for like 2 hours but one glance at the diagram made it clear

[@olefrehr]

WOW! You can not beat that handwriting! So c crisp and clear. Thank you for sharing.

[@georgepolasky9809]

Dear Krista, I have faithfully watched yours and Christine Breiner’s (of MIT Courseware) Calculus videos. Truthfully, I am beyond amazed at both of your levels of skill and brilliance, and the ability to teach and do so in such a pleasant, down to earth, caring way. Thank you both for all you do and please continue with more videos. I am nowhere near your levels, but I do learn from you, and it is a very enjoyable learning experience each time. Blessings, George.

[@kristakingmath]

Thank you so much, George! I'm so happy that you enjoy the videos and are able to get great use out of them... I really appreciate you saying hi and letting me know! :)

[@Iamhuman03]

@@kristakingmath Krista mam I also enjoyed 😊

[@SidviciousWisconsin]

Thank you, old guy going back to school needs all the help he can !

[@kristakingmath]

Good for you for going back, that's awesome! :)

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[@ValiChandiwala]

Been trying to search for this for 2 weeks. Finally understood. I remembered studying this in engineering some 13-14 years ago but I couldn't remember why the oval shape comes in and why it's slanted this way instead of that way. Danke!

[@Memes_uploader]

the best explanation I have ever seen

[@chidinmaadogu8242]

Best video I’ve found on this literally!!!! Thank you

[@sistersfusion2708]

I love your videos! They helped me a lot through my calc 3 course.

[@kristakingmath]

Thank you so much, I'm so glad I could help! :)

[@sistersfusion2708]

Thank you! please, keep uploading more videos ))

[@braulioeducator]

Great video Krista King!

[@kristakingmath]

Thank you so much!! :)

[@mousaalsaeed9410]

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[@CSGdewxp]

Great video. Great drawings. Great Explanations. SO simple. Thank you

[@atika97.]

omg !! the bestest video ever n thanks 😊

[@chebetmark6724]

Well-done, understood

[@jahansaid6382]

Krista King, I gotta admit you are good, and very clear at your explanation. I watch a couple of your video, and now I can recognize the quadratic surface like drinking water as easy as peace of cake.

[@kristakingmath]

Oh good! I'm so glad the videos have been helping! :D

[@rein29233]

It was very easy understand to with your explanation. Thank u very much.

[@kristakingmath]

You're very welcome, I'm glad it made sense! :)

[@GiI11]

Thank you! That was great review.

[@ayknsancakl2983]

Clear and great explanation ! thank you so much.

[@kristakingmath]

Thank you! I'm so glad it helped! :)

[@sameeramjad1598]

THANK YOU SO MUCH SHERA

[@kristakingmath]

You’re so welcome!! ❤️

[@fggcreed]

I wish I'd seen this before my first test >.

[@prasoonsharma87]

Really nice video..... Easy to understand!!

[@rubengomez8780]

Omg, your videos are absolutely great, thank you very much!

[@kristakingmath]

I'm so glad they're helping! 😊

[@manmeetgulati6208]

Thank you so much........... Best explanation. I was confused a lot before watching this video... Now the things are more clear to me

[@kristakingmath]

You're welcome, Manmeet! I'm so glad it helped! :D

[@parkjessica4444]

This is so helpful! Love your clear explanation and voice!!!thank you

[@kristakingmath]

You're welcome, Jessica, I'm so glad this helped! :)

[@lesliederrar9718]

thank you

[@StaticSleet]

Added to my education playlist. I only save the best on youtube. Thank you...and my future self who will also probably use this to relearn thanks you as well

[@Max-dn1sj]

I like you and like your video explaining the level curves

[@bowenyi1833]

Thanks for your amazing explanation!

[@kristakingmath]

You're welcome, Bowen, I'm so glad you liked it! :)

[@cabrakan0883]

Your videos are amazing. Thank you so much!

[@michaeldean4362]

Extremely helpful!! Thank you!

[@kristakingmath]

You're welcome, Michael! I'm so glad it helped! :D

[@danteeep]

It all makes sense now thanks a lot

[@abdelz1617]

Great video!

[@kristakingmath]

Thanks, abdel! :)

[@Com-pf3ij]

Super explanation ...ma'am ..1👌👌

[@kristakingmath]

Thanks a lot, papa! :)

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omg you the best saved my life and time

[@kristakingmath]

I'm so glad it helped, amir! :)

[@unnatishukla1596]

Helpful really!

[@kristakingmath]

Glad it was helpful, Unnati! :)

[@drewpierpont3361]

This is the first video I've ever seen offered at 4K/60fps

[@nothandosithole7586]

that was great,loved your explanation,thank you.

[@Lostwolf16]

Best explanation with 4K video means I can skip the class and still get an A. Thank you so much

[@kristakingmath]

No problem, Prabin, glad it helped! 👍

[@wahidnoori2259]

Thanks, very helpful to me.

[@kristakingmath]

You're welcome, Wahid! I'm so glad it helped! :)

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Great video! well explained.

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اعظم مدرسة ❤️❤️😘

[@LittleWaffle]

Very helpful, thanks a lot!

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i like your video. thanku😍

[@tierraterry8451]

Thank you!

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best explanation ever I got 😘😘😘

[@prakhart92]

Awesome...❤️

[@ahmeduzzaman4704]

Life saver.💝

[@rezaetemad591]

great video ...just one thing...if you replace the C in the second example with a constant, then you are going to get a Hyperbola instead of an ellipse. but over all well explained. thank you.

[@solinsaadi2473]

thank you 😍

[@calmspace8221]

Can you sketch level curves of z=1/xy.

[@arnomarsh8018]

May I be so bold and ask what program/hardware you use to write these lessons? Thanks for the awesome content!!!

[@kristakingmath]

Hi Arno! I explain here: www.kristakingmath.com/blog/how-i-create-my-videos

[@rabeash711]

👏👏👏Thank you alot

[@MrRynRules]

thank you!

[@kristakingmath]

You're welcome! :)

[@shobhit7212]

Difficult concept with easiest way.....Thank a lot.

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[@MRAXELGRINDER]

Awesome

[@chemist7908]

Perfect

[@theonlythingihavetosayis9333]

What if you dont realize the inherent curve of a function. Like knowing the equatuin of an eclipse for the second example? All these example on youtube are just for recognized equations

[@lukschs1]

TE AMO

[@jordanwrong2042]

you are a gift from god

[@Justinian21c]

Could you tell me please why the ellipsoid in the second example is elongated along the x-axis? The x^2 is divided by 4 but y^2 isn't, so I don't understand why x would be elongated. It seems to me that x would be shrunken more if divided by 4. Thanks.

[@cyberprinc3ssASMR]

with the ellipsoid equation, the denominator of x and y correspond to the 'magnitude' of the ellipsoid. meaning that, if you have (x^2 /a^2) + (y^/b^2), the size of the elipsoid will be: a (in x axis) and b (in y axis). In the example, x is divided by 4, (therefore the elongation will be in 2 **because root of 4 = 2** ), and y is divided by 1 (meaning its elongation in the y axis will be of 1). if you plot the points in the 2d graph, youll see that x is elongated

[@maxpayne5257]

God bless you

[@jakewar7201]

Hi thanks a lot for your amazing videos If I want to study calculus 3 from your videos what is the order of the tutorial

[@kristakingmath]

Hey, Jake! Focus on Partial Derivatives first, then Multiple Integrals, then Vectors. :)

[@alyssameylsaechao]

What program do you use for your sketches and videos? Its awesome!

[@kristakingmath]

Hey Alyssa! It's called Sketchbook. :)

[@vrendus522]

Numerology about Lahinch Numerology (Expression Number), correct?

[@ferry6700]

I love your voice!

[@anlyldrm2714]

How can i sketch the level curves when the question contains sin,cos,tan and others ?

[@manjeet1278]

mam,can u post funcuntion video for beginers. ☺

[@SachchidanandPrasad1729]

Can you please tell me which software are you using? It will really help me to teach the students in the class

[@kristakingmath]

It's called Sketchbook. If you want more information, here's how I put the videos together: www.kristakingmath.com/blog/how-i-create-my-videos

[@SachchidanandPrasad1729]

Okay, thanks a lot.

[@SachchidanandPrasad1729]

Thanks for it. Did you graph also in the sketchbook?

[@gavingoh2501]

May I know what apps do you use to do such things?

[@kristakingmath]

I use Sketchbook and Sceenflow, along with a Wacom pen tablet. :)

[@zachadams7748]

Does anyone know the name of the software used in this video?

[@kristakingmath]

Hey, Zach! I use Sketchbook (by Autodesk) for the blackboard, with an old Bamboo tablet from Wacom, and then Screenflow to record and edit. :)

[@michaelquinatoa139]

Nicely explained, just please label your axes :D especially in the 3D plots

[@shawnscientifica6689]

Krista is clearly a goddess and we are lucky for her presence

[@ChanchalDass]

Three-dimensional graph of z=x^2+y^2 is not correct

[@kagayakuangel5828]

died at 9:10 lmaoooooo thank you

[@lalitakaushik3862]

you look like sehnaz

[@carrilpointdujour2530]

Started wondering about orbits, gravity, and excuse me boobs

[@vijayprime9693]

thank you