Louis

Been working to better understand calculus through youtube videos, textbooks and the like and came across this when looking into differentiability involving multivarable functions, specifically in this video around 32:00

https://youtu.be/CNKk3qFswp0

I thought to be differentiable the function had to exist at the point in question, and here (0,0) would make it not exist. Unless I am wrong about that.

So I tried finding the radius, got sqrt(2)/2, so as far as I can tell It's the circle with the equation (x - .5)^2 + (y - .5)^2 = .5. The two points are exactly pi radians from each other in reference to the center, so it's half the circle. I went ahead found the circumference 2*pi*sqrt(2)/2 = pi*sqrt(2) then divided it by two so it's half the circle and got pi*sqrt(2)/2.

The book says it should be pi/2, so more just verifying if I'm right.

So basically something with no rational roots. Is thisa problem that would be given at or just below a calculus level?

For a more solid example, say someone is on top of a hill and another is on the slope of a hill and they are each looking at an object in the air 3 meters in front of them and four meters in the air, so there are two different objects. Would the angle of elevation for one be different than the other?

I guess part of the question is where the horizontal is determined.

Question came to mind after I was asked if x^2 + xy = 1 had any symmetry. Was able to tell it had symmetry about the origin, but looking at the graph it looks like it has symmetry about some other line with a negative slope.

Even if I'm mistaken about this, I am still wondering if there are ways to tell if other symmetries exist mathematically similarly to the other examples.

On khan academy, an article mentions finding the 2d curl is basically taking the x and y component with the two partials, or at least that's how I understand it, so its gives twice the magnitude.

Later on in the article for a 3D curl it says you still divide it by 2, wouldn't you divide it by 3? At any rate here's the article.

https://www.khanacademy.org/math/multivariable-cal...

How would I go about solving this? each letter represents a digit, and E, A and W are in the ones place.

I can tell 1+A= T, which means Tand H add together to make 10+E, but it could also be 1+T+H from carry over from the 100s place, and from there it just gets more complicated from how I'm looking at this. ​

https://us-static.z-dn.net/files/d04/fff0bebc85d91...

Hard to explain, and maybe I'm not explaining it correctly or fully understanding what's going on. But shown in this khan academy video around 7:00 the surface holding up the 5 N object has a normal force acting downward.

https://youtu.be/MaabUHLIIXA?t=419

As I understand it the normal force is an object "pushing back", keeping say something resting on top of it from pushing through. Ensuring there i no net force making an object move.

A normal force pointing downward then means there is something pulling upward? Really not sure here. Thank all in advance for any help

I hope this is an alright place to ask this. I'm on han academy, on the linear algebra class and get to this video.

https://youtu.be/eEUK_ThrHuQ

2 minutes in or so and I'm absolutely stuck. Not sure what's going on with the vector [c1, c2], but I assume it doesn't have to do with the c in the matrix?

It looks like it is saying a transformation on that vector, which results in the new vector [(c1), (ac2 - cc1)] yes? In which case the transformation matrix is a 2x2 that looks like [(1, 0), (a, -c)] yes?

Trying to take this step by step to figure it out on my own, also not so sure how the following matrix is gotten , though maybe I will once I understand what happened int he beginning.

I may be misunderstanding the concepts of domain, range, codomain, image and preimage, and that may be contributing to my misunderstanding of the surjective functions.

As I understand The domain is all applicable elements a function can use, so it doesn't have to be all real numbers.

The codomain is all reals (when dealing with simple functions f(x)=y) then the image/ range as subsets of the codomain.

The image is the subset of the codomain that a subset of the domain is mapped to, and if the subset of the domain is the whole set then it is called the range.

The preimage is elements of the domain that map to a chosen subset of the codomain, though not all elements of the codomain need to be mapped to.

Admittedly the preimage doesn't matter too much with my question, though if it or any of what i listed as my understanding is incorrect, do let me know.

Also kind of curious if there is a similar concept to the codomain for the domain, where the domain is a subset of it.

For my question though, the definition for surjectivity seems to be when the codomain and range are the same. Maybe a better question then is why does the codomain need to be all reals? Or could you specifically say a function that maps from the domain to a set of elements that make up the range and have that be surjective?