I feel discouraged to start studying with Leonard because of his lengthy videos. Soon as I press play and hear him speak though, it's like a caffeine shot through my mind. He's so good at keeping us focused and confident.
I'm starting a PhD in pure math cause of how much this man has helped me, still go back to his videos from time to time to see someone who truly loves mathematics
0:00 Explanation of vector fields 6:37 Note for graphing vector fields 8:45 Example 1 (constant vector) 11:40 Example 2 18:30 Example 3 (vectors tangent to circle) 29:15 Example 4 33:50 Example 5 39:06 Extrapolations about vector fields (finding commonalities among different functions) 41:10 Example 6 (R-3 vector field) 44:50 Calculus of vector fields (gradient) 48:50 Conservative vector fields 54:10 Example (finding conservative vector field)
He is beloved because we have all felt it, that complete feeling of inadequacy. That, other people are understanding in class but we aren't. That, maybe we should be rethinking our career goals and majors entirely, because we just don't understand no matter how many questions we ask or how many times we read the book -- it just doesn't click.. Professor Leonard has a gift. I am just personally so grateful that he shares it for free to everyone who needs it. He has armed me with more than just math knowledge, but also courage and hope.
Almost the same. In Calc 1 and some of calc 2, I had no idea what I was doing and ended up with C's or B's. Now in calc 3, I watch Professor Leonard's videos as a supplement to pretty much every topic we cover in class, and I have yet to fall under a 95 on a test. It's astonishing how much of a difference the quality of a teacher can make, not only on grades, but on genuine understanding.
Hey Professor Leonard, I just wanted to express my gratitude. This semester with covid and all, I had to take multivariable calc online. I have not heard a word of instruction from my professor--he just directs us to videos from awesome people like you. What my prof should be doing (because he is getting paid), you are doing for FREE. Between calc 3 and diff eq's, you've made this crazy semester so much more manageable. Thank you so much.
Professor, I greatly appreciate your videos. I am a mechanical engineering student in Argentina, and everything you gave throughout your calculus 3 course has helped me. I have watched each of your videos starting from vector functions. Thank you, thank you, thank you, you are a great teacher. I congratulate you for your work.
I wish I had had access to this lecture when I was an undergraduate. He is a brilliant and committed teacher. Other teachers could really learn from him.
Oh my god, I need to get through these videos quickly and I thought 1.25 speed would be too fast but I just tried this and it's actually working so well for me, especially since he's not speaking when he's writing. Thanks for the tip!
1.25, 1.5 nah 2x speed is the way to go. The only videos I watch on youtube not on 2x speed is music videos.The odd time if the youtuber already talks very fast or has a strong accent maybe 1.5 but that rarely happens. To be honest I want a faster mode 2.5 maybe 3x.
My teacher, for the first time in my life... is actually horrible. Watching your videos makes up COMPLETELY for what I've been missing this semester. THANK YOU SO MUCH!!!!!
I was wondering why these vids help me so much when my lecturer also explains stuff like this. I realized it's coz Leonard asks questions constantly and it makes your brain stay connected to what he says. Even if it's a really obvious question it's like he ensures everyone's holding on to a thread of what he's sayin and nobody is left behind. thanks bro
I wish I had had access to this lecture when I was 2 year undergraduate. He is a brilliant and committed teacher. Other teachers could really learn from him.
Professor Leonard,thank you for a solid Introduction to Vector Fields and what makes them conservative in Multivariable Calculus. I also encountered this material in Engineering Physics One and Engineering Electromagnetics. This material is simple to follow/comprehend in Calculus with examples, problems and graphs.
I guess what isn't obvious to you though is sarcastic rhetoric! But yeah, this guy is pretty damn cool. I found myself uncontrollably smiling when things he said "clicked" in my brain and I could actually feel each time I "learned" something. I mean, I've been out of school for 10 years now, and I can say that I know how to do vector fields in calculus! (again?)
For my first exam, watched all his videos, went through all my homework and class examples and still got a 39. My teacher is bad at picking test problems...
Thank you. I love your teaching and your genuine care for those who do not seem to understand. I finally understand. Sometimes its not you as a student, but the professor that really screws up. I have come to love you after one video.
I have a math degree and did well in calc3, but I didn’t really understand the core concepts the way I did with calc 1/2. Vector fields are where it really went downhill for me and these videos are filling that gap in my understanding. Also, a lot of these concepts are important for understanding complex analysis so it is also helping me with my understanding of that subject. Thanks a bunch!
OMG! goose bumps at 48.! wowee. this is what I study for! A gradient IS a ... (don't want to ruin the suspense). Thanks forever Professor S. "Souperman" Leonard! I can't wait for the next one when you let us in on the 'curl' and 'divergence' secrets! wowee!
plug a point in a function and draw a vector from that point. repeat this for multiple points in the cartesian plane and you'll see a pattern gradient vector of a function f is a conservative vector field. f becomes the potential function 50:25. vector field is conservative if it's equal to gradient of some function (called the potential function)
you really make me feel mathematics continue work like that dear sir I do thank God located me your channel even if there is no teacher I understand it
My calc 3 class has been perfectly in sync with these video lectures up until this point. For some reason, my instructor is including differential equations for the potential function exercise at the very end of this video. We are having to calculate the potential function using the F(x,y,z), treating it as a gradient and integrating its components to end up with a differential equation that needs to account for a possible constant.
Dear Prof, How would you graphically demonstrate a conservative vector field? What is its physical meaning? Can a potential function from which we are evaluating conservative field be itself a vector field or not? What do potential function and conservative vector field together tells us? I shall be grateful for answering this question?
To graphically demonstrate a conservative vector field, what you look for is whether there is any rotationality to what the field could cause. In other words, imagine that the field represents a force at each point in space. Is there a preferred path you could navigate through the field in a closed loop, and have the field speed you up? If yes, then it is not a conservative field. If there is an advantage to letting the field do work on you as you circle clockwise, instead of CCW, or vice versa, then the field is non-conservative. If a vector field were to represent a fluid's velocity, then you would look for whether the vector field could cause a pinwheel to rotate. A conservative vector field would keep an initially stationary pinwheel in its original orientation, as it is cast along with the flow of the fluid. Uniform vector fields are conservative. Coulomb's law and Newton's law of gravitation generate conservative fields for electrostatics and gravitation respectively.
A potential function is not a vector field. It is a scalar function. By definition, a conservative vector field is the gradient of a scalar function. You cannot take the gradient of a vector field, since gradients are only applicable to starting with scalar fields (a process analogous to scalar multiplication). The potential function is a shortcut to calculating the line integral between any two points in a vector field, such that you can simply evaluate the potential function at the start and end points, instead of setting up the line integral. An application of this, is when we use potential energy functions of electric and gravitational fields, to evaluate work done between two points as an object moves through the field. This only works for conservative fields. Magnetism can make electric fields non-conservative, so it gets much more complicated to evaluate work done by an electric field in that context. Also, given a potential function, the gradient vector field of it, that is conservative, will tell you the direction of steepest ascent at every point in space, and what that directional derivative is. Picture the potential function as telling you the topographical height of a landform like a hill. The gradient tells you the steepest path to climb, and what slope you would have to climb. It doesn't necessarily need to work this way, since there are other applications of this concept. Another example would be a potential function representing how temperature varies throughout a region of space. The gradient would tell you the direction of heat flow, and would be proportional to the rate heat flows based on the conductivity of the material as well.
I am bordering on a B+/A- in Multivariable Calculus. Vector fields is the last chapter. I hoping I may pull off an A- but would still be very happy with a good solid B. Thank you for the vids.
no, every vector in this vector field should have the length of 1 because if you add the two components as Pythagoras theorem you always get 1, this is the point of choosing this vector field
AxanLderE the problem is that there is no any vector in this vector field with the coordinates of ( 1,2) according to the definition of that vector field he ia trying to represent now the vector ahould have the coordinates of ( 1/sqrt(5) , 2/sqrt(5) ) am I wrong until here ? but he drew (1,2) instead !!
i am a bit cofused with the third vector field example after the 20th minute. When we have the function y for i direction why we use the final value for i to move along the x direction?
Hello! I am taking Calculus 3 online and your videos have been sure helpful! I was just curious since there is a different book that my class is using, or should I say a newer version, I was just wondering if there is anything new over the last 4 years that is not covered in these lectures for this chapter?
+professor LEONARD great jobbbb you are beyond amazing actually, I am still trying to just one section that you are not covered in your lessons and that is supremum and infimum of function its really hard
Because f(x,y) spits out a 3rd value as the Z component, and F(x,y) as a vector field spits out a 2 dimensional vector associated with the point in 2D space that was plugged into the vector field. A 3D vector field would have 3 inputs; which are associated with a point in 3D space and outputs a 3 dimensional vector associated with the point in 3D space; whereas, with a 3 variable function, the output would be in the 4th dimension.
Professor, can we find the respective potential function from a given conservative vector field? is there any method to do that?? And nice lecture, professor.
Yes. I'll give you an example. Consider F = . You can trust me, or calculate the curl if you'd like to confirm it is conservative. Integrate the first function relative to x, and set up an arbitrary function of y and z as the constant: int y dx = x*y + g(y, z) Integrate the second function relative to y, and set up an arbitrary function of x and z as the constant: int (x + 2*y) dy = x*y + y^2 + h(x, z) Integrate the thrid function relative to z, and set up an arbitrary function of x and y as the constant: int 2*z dz = z^2 + j(x, y) Now reconcile these three results, to fill in our functions g, h, and j. Notice that x*y appears in the first two, which helps us reconcile. This means g(y, z) would equal y^2 + h(x, z). We now have: x*y + y^2 + h(z) And the third result is only a function of z, which is a perfect fit for h(z). The only thing left to do, is add the arbitrary constant K. Conclusion for function f, such that F=grad f, is: f(x, y, z) = x*y + y^2 + z^2 + K
I feel discouraged to start studying with Leonard because of his lengthy videos. Soon as I press play and hear him speak though, it's like a caffeine shot through my mind. He's so good at keeping us focused and confident.
was feeling the same way 1h ago. now i dont wanna cry anymore😂😂
first test without leonard: 0%
second test with leonard: 100%
gym time: increased 110%
AAHHAHAHAHA
This is so real. #Gains
I literally passed my vector calculus class with A just because of these videos. They are gems.
Great Job!!
I'm starting a PhD in pure math cause of how much this man has helped me, still go back to his videos from time to time to see someone who truly loves mathematics
Thats so inspiring , good luck
hows it goin
How's your math PhD going? I'm genuinely curious
0:00 Vector Field basics
44:50 Gradients and their relationship with CONSERVATIVE vector fields
0:00 Explanation of vector fields
6:37 Note for graphing vector fields
8:45 Example 1 (constant vector)
11:40 Example 2
18:30 Example 3 (vectors tangent to circle)
29:15 Example 4
33:50 Example 5
39:06 Extrapolations about vector fields (finding commonalities among different functions)
41:10 Example 6 (R-3 vector field)
44:50 Calculus of vector fields (gradient)
48:50 Conservative vector fields
54:10 Example (finding conservative vector field)
He is beloved because we have all felt it, that complete feeling of inadequacy. That, other people are understanding in class but we aren't. That, maybe we should be rethinking our career goals and majors entirely, because we just don't understand no matter how many questions we ask or how many times we read the book -- it just doesn't click..
Professor Leonard has a gift. I am just personally so grateful that he shares it for free to everyone who needs it. He has armed me with more than just math knowledge, but also courage and hope.
Give this man a medal.
Indeed.
But he didn't explain that R-3 vector fields are just a 6-dimensional space, with 3 upper dimensions projected onto the lower 3. No medal.
Honestly
Test without lenoard: 42%
Test with Leonard: 92%
Read this comment whenever you feel down
ive scored 90% + on all exams since using Prof Leonard. This man needs a RAISE!!!!
I aced my exams with him, after averaging 60% in cal 1
SaceedAbul same: first test without Leonard 75. first test with Leonard 95.
Update on my situation: got 95 in my cal 3 final and I never went to class only watched all of leonard s videos the class avg was 70 thank you
Almost the same. In Calc 1 and some of calc 2, I had no idea what I was doing and ended up with C's or B's. Now in calc 3, I watch Professor Leonard's videos as a supplement to pretty much every topic we cover in class, and I have yet to fall under a 95 on a test. It's astonishing how much of a difference the quality of a teacher can make, not only on grades, but on genuine understanding.
I don't know where I'd be without this videos, Must say your students has got the best lecturer of all times.
Thank you very much prof!
Hey Professor Leonard,
I just wanted to express my gratitude. This semester with covid and all, I had to take multivariable calc online. I have not heard a word of instruction from my professor--he just directs us to videos from awesome people like you. What my prof should be doing (because he is getting paid), you are doing for FREE. Between calc 3 and diff eq's, you've made this crazy semester so much more manageable. Thank you so much.
Professor, I greatly appreciate your videos. I am a mechanical engineering student in Argentina, and everything you gave throughout your calculus 3 course has helped me. I have watched each of your videos starting from vector functions. Thank you, thank you, thank you, you are a great teacher. I congratulate you for your work.
I wish I had had access to this lecture when I was an undergraduate. He is a brilliant and committed teacher. Other teachers could really learn from him.
from overseas Saudi Arabia I am studying physics whenever I get stuck in physical calculus I come here to see this great man. that's a bless
"Don't go to bed with a snake by your head."
-Professor Leonard
Fantastic lecture as well.
You and PatrickJMT are a godsend. Thank you for all the help!
They're like Superman and Ironman
Professor Leonard is a god sent. Thank you for increasing my confidence in my calculus course.
NO One explained Vector fields and gradient like this. You've given me new eyes.
Idk if these students know how lucky they are to have such an engaging teacher especially in calculus
So I watched this lecture at 1.25 speed and in 46 minutes you explained it better than a 2 hour lecture.
Oh my god, I need to get through these videos quickly and I thought 1.25 speed would be too fast but I just tried this and it's actually working so well for me, especially since he's not speaking when he's writing. Thanks for the tip!
Same.
fucken genius
i do it at 1.5x speed. still completely understandable. cuts the time down super fast.
1.25, 1.5 nah 2x speed is the way to go. The only videos I watch on youtube not on 2x speed is music videos.The odd time if the youtuber already talks very fast or has a strong accent maybe 1.5 but that rarely happens. To be honest I want a faster mode 2.5 maybe 3x.
Professor Leonard the God of maths. He is the hero we students need and 100% the one we deserve
You are my inspiration to study CALCULUS. You made it so easy to understand.
Thank you so much Professor Leonard
Love from Nepal 🇳🇵🇳🇵🇳🇵
Oh my God!!!! It feels like he cares about explaining!!! So rare... i loved how you teach! #CountryOfPlastic #Educational #Recommendation #Math
My teacher, for the first time in my life... is actually horrible. Watching your videos makes up COMPLETELY for what I've been missing this semester. THANK YOU SO MUCH!!!!!
I was wondering why these vids help me so much when my lecturer also explains stuff like this. I realized it's coz Leonard asks questions constantly and it makes your brain stay connected to what he says. Even if it's a really obvious question it's like he ensures everyone's holding on to a thread of what he's sayin and nobody is left behind. thanks bro
Thank you for being the standard that other professors should be striving to be.
Best teacher I've ever had. Thank you for videos. They are the only reason I get the material. Thanks!
I wish I had had access to this lecture when I was 2 year undergraduate. He is a brilliant and committed teacher. Other teachers could really learn from him.
Hands down the best math teacher to ever walk this earth! Can't thank you enough for being our Clark Kent, professor!
Professor Leonard,thank you for a solid Introduction to Vector Fields and what makes them conservative in Multivariable Calculus. I also encountered this material in Engineering Physics One and Engineering Electromagnetics. This material is simple to follow/comprehend in Calculus with examples, problems and graphs.
it is possible to fail a test when you have such a teacher ? :)
Thanks a lot Professor Leonard
Obviously
I guess what isn't obvious to you though is sarcastic rhetoric!
But yeah, this guy is pretty damn cool. I found myself uncontrollably smiling when things he said "clicked" in my brain and I could actually feel each time I "learned" something.
I mean, I've been out of school for 10 years now, and I can say that I know how to do vector fields in calculus! (again?)
For my first exam, watched all his videos, went through all my homework and class examples and still got a 39. My teacher is bad at picking test problems...
I just wanna let you know that I'm grateful for you. You're a lifesaver
Thank you. I love your teaching and your genuine care for those who do not seem to understand. I finally understand. Sometimes its not you as a student, but the professor that really screws up. I have come to love you after one video.
I have a math degree and did well in calc3, but I didn’t really understand the core concepts the way I did with calc 1/2. Vector fields are where it really went downhill for me and these videos are filling that gap in my understanding. Also, a lot of these concepts are important for understanding complex analysis so it is also helping me with my understanding of that subject. Thanks a bunch!
OMG! goose bumps at 48.! wowee. this is what I study for! A gradient IS a ... (don't want to ruin the suspense). Thanks forever Professor S. "Souperman" Leonard! I can't wait for the next one when you let us in on the 'curl' and 'divergence' secrets! wowee!
plug a point in a function and draw a vector from that point. repeat this for multiple points in the cartesian plane and you'll see a pattern
gradient vector of a function f is a conservative vector field. f becomes the potential function
50:25. vector field is conservative if it's equal to gradient of some function (called the potential function)
Conservative Vector field ~ 45:02
Even with a good calculus professor like mine, these are still nice to watch as refreshers;especially while I'm reviewing for my final.
locking in for the final. the prof leo calculus journey was legendary. ill be back for diff eq, then that’s probably it :(thank you)
you really make me feel mathematics continue work like that dear sir I do thank God located me your channel even if there is no teacher I understand it
Anyone aspiring to be a teacher/professor should look up to Professor Leonard. Thank you, Professor!
My great professor confused me so much with all the formulas I didn't know what is what! after watching you I got an 89! Thanks!
Natural talent for teaching
The gradient of a function is a specific type of vector field called a Conservative Vector Field. This is explained more thoroughly starting @ 49:19
Excellent Math teacher with an excellent physique, first time i listen to him, and i never seen a fit math teacher, lol.
I never had a such a good prof.I wish I am able to understand maths
i dont believe this is free, he has passion within his teaching and deep understanding , what a proffessional
i never turn off my adblocker, but when i do, i watch this channel.
Me distracted, looking at my phone
13:20 "Lemme have your eyes up here"
o_o
In my university, class called math 3 starts from this vid. It is going to be my last math class in Uni. I really hope i pass this course.
You will go down
as a legend leonard
So why aren't there any Liberal vector fields? : P
Cause theyre all still crying after loosing to the conservatives :p
because Conservative is superior
MickyR Go fuck yourself you pice of trash
Remind me again why we have to politicize calculus? smh
@@ShreyButle please smh
Calc 3
test without leonard 57%
test with leonard videos 96%
no joke
love the energy, love the enthusiasm
I wish all professors were as amazing as Professor Leonard because I swear I could learn anything
You are such a god Leonard hahaha TRULY!!!! Watched it at 2 speed and you were hella understandable!
Thank you for this !
I haven't had this much fun since connect-the-dots!
My calc 3 class has been perfectly in sync with these video lectures up until this point. For some reason, my instructor is including differential equations for the potential function exercise at the very end of this video. We are having to calculate the potential function using the F(x,y,z), treating it as a gradient and integrating its components to end up with a differential equation that needs to account for a possible constant.
Dear Prof, How would you graphically demonstrate a conservative vector field? What is its physical meaning?
Can a potential function from which we are evaluating conservative field be itself a vector field or not?
What do potential function and conservative vector field together tells us?
I shall be grateful for answering this question?
To graphically demonstrate a conservative vector field, what you look for is whether there is any rotationality to what the field could cause. In other words, imagine that the field represents a force at each point in space. Is there a preferred path you could navigate through the field in a closed loop, and have the field speed you up? If yes, then it is not a conservative field. If there is an advantage to letting the field do work on you as you circle clockwise, instead of CCW, or vice versa, then the field is non-conservative.
If a vector field were to represent a fluid's velocity, then you would look for whether the vector field could cause a pinwheel to rotate. A conservative vector field would keep an initially stationary pinwheel in its original orientation, as it is cast along with the flow of the fluid.
Uniform vector fields are conservative. Coulomb's law and Newton's law of gravitation generate conservative fields for electrostatics and gravitation respectively.
A potential function is not a vector field. It is a scalar function. By definition, a conservative vector field is the gradient of a scalar function. You cannot take the gradient of a vector field, since gradients are only applicable to starting with scalar fields (a process analogous to scalar multiplication).
The potential function is a shortcut to calculating the line integral between any two points in a vector field, such that you can simply evaluate the potential function at the start and end points, instead of setting up the line integral. An application of this, is when we use potential energy functions of electric and gravitational fields, to evaluate work done between two points as an object moves through the field. This only works for conservative fields. Magnetism can make electric fields non-conservative, so it gets much more complicated to evaluate work done by an electric field in that context.
Also, given a potential function, the gradient vector field of it, that is conservative, will tell you the direction of steepest ascent at every point in space, and what that directional derivative is. Picture the potential function as telling you the topographical height of a landform like a hill. The gradient tells you the steepest path to climb, and what slope you would have to climb. It doesn't necessarily need to work this way, since there are other applications of this concept. Another example would be a potential function representing how temperature varies throughout a region of space. The gradient would tell you the direction of heat flow, and would be proportional to the rate heat flows based on the conductivity of the material as well.
I am bordering on a B+/A- in Multivariable Calculus. Vector fields is the last chapter. I hoping I may pull off an A- but would still be very happy with a good solid B. Thank you for the vids.
37:05
Why is the vector at (2,1) longer than the others? I think it has to be as long as the rest!
He kind of explains at 36:45
The function is telling us to make the distance traveled along X to be twice the length of the Y.
no, every vector in this vector field should have the length of 1
because if you add the two components as Pythagoras theorem you always get 1, this is the point of choosing this vector field
AxanLderE the problem is that there is no any vector in this vector field with the coordinates of ( 1,2)
according to the definition of that vector field he ia trying to represent now the vector ahould have the coordinates of ( 1/sqrt(5) , 2/sqrt(5) )
am I wrong until here ?
but he drew (1,2) instead !!
Yeah you got it, he mistakenly graphed it as 2i + j instead of 2/sqrt(5)i + 1/sqrt(5)j.
It's not to scale.
This video is brilliant.
Props to this guy!
I just discovered this amazing teacher on the day of my final :(
same, it's ok. watch them after the final is still worth it
i am a bit cofused with the third vector field example after the 20th minute. When we have the function y for i direction why we use the final value for i to move along the x direction?
Conservative Vector Field at 49:35
Thank you!
Good looks
Bless you sir
god bless
At 11:00 does the map have to go from R2 to R2 or could we have R2 to R3. Likewise could we have a map from R3 to R2 or R1 etc.
if that channel did not exist I would nvr pass my calc classes
Sir you are a real super Hero of math
Thank you very much Professor Leonard
Thank you Mr professor..:)....you are really doing an awesome job...:)
The students in his class are blessed
Johnny Bravo, thank you.
very good teacher:) and you talk smooth
Very clearly explained fom the basics
stellar explanation
34:00 Example
Thank you for posting this good lecture. Finally got it!
Its so nice of Superman to take time out of his crime fighting schedule to teach me math!
Hello! I am taking Calculus 3 online and your videos have been sure helpful! I was just curious since there is a different book that my class is using, or should I say a newer version, I was just wondering if there is anything new over the last 4 years that is not covered in these lectures for this chapter?
Professor Leonard is the best, I want to have a drink with him.
"Don't go to bed with a snake by your head," says perhaps the greatest math teacher of all time.
He is really awesome teacher
Thank you Professor Leonard
+professor LEONARD great jobbbb you are beyond amazing
actually, I am still trying to just one section that you are not covered in your lessons
and that is supremum and infimum of function
its really hard
I’m confused. Why does f(x, y) relate to a vector function in 2D when f(x, y) has been in 3D prior to vector fields?
Because f(x,y) spits out a 3rd value as the Z component, and F(x,y) as a vector field spits out a 2 dimensional vector associated with the point in 2D space that was plugged into the vector field. A 3D vector field would have 3 inputs; which are associated with a point in 3D space and outputs a 3 dimensional vector associated with the point in 3D space; whereas, with a 3 variable function, the output would be in the 4th dimension.
Thank you for your "super power" sir !
Would it be correct to say all vector fields are conservative fields?
How can I thank you prof leonard for your awesome videos?
thank you leonard
its very good
i live you form iran
Awesome lectures!!!
Where'd the arms go? :(
Why do Vector Fields start at the point given ant not (0,0)
Great explanation! Thank you!!!!
Professor, can we find the respective potential function from a given conservative vector field? is there any method to do that??
And nice lecture, professor.
Yes. I'll give you an example.
Consider F = . You can trust me, or calculate the curl if you'd like to confirm it is conservative.
Integrate the first function relative to x, and set up an arbitrary function of y and z as the constant:
int y dx = x*y + g(y, z)
Integrate the second function relative to y, and set up an arbitrary function of x and z as the constant:
int (x + 2*y) dy = x*y + y^2 + h(x, z)
Integrate the thrid function relative to z, and set up an arbitrary function of x and y as the constant:
int 2*z dz = z^2 + j(x, y)
Now reconcile these three results, to fill in our functions g, h, and j. Notice that x*y appears in the first two, which helps us reconcile. This means g(y, z) would equal y^2 + h(x, z). We now have:
x*y + y^2 + h(z)
And the third result is only a function of z, which is a perfect fit for h(z). The only thing left to do, is add the arbitrary constant K.
Conclusion for function f, such that F=grad f, is:
f(x, y, z) = x*y + y^2 + z^2 + K
Thank you, Professor Gains
its possible to understand at 1.5 speed and its the quickest I've gone from not understanding anything about vector fields to mastering the basics
you know you're a good teacher when your math videos have 150k views
Please can you make a videos on Differential Equations both ordinary and partial.
Very concise. Thanks for the instruction.
in the last example
the initial condition x + z > 0 must be set