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IA : VECTOR FUNCTIONS


chumbuks

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Hey guys,

So for my HL Math IA, I stumbled upon vector functions after looking at one that suggested cycloids, which led to brachistrone curve, I'm not 100 sure as I didnt read much up on brachistrone curve itself. So I liked looking into vector functions, and looking into the calculus with vector functions. So I was wondering if anyone had any suggestions on what I could investigate more specifically using vector functions and calculus of vector functions

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Depends on how much you want to learn or what you're interested in. I just took the course so if you need any help with anything, let me know.

Vector functions relate a lot to physics. For example, you can use a vector function to parametrize the trajectory of a particle (notably when the path is parabolic). You can use a line integral over a vector field to find the force done by a particle traversing the field. You can then go into conservative vector fields and the fundamental theorem for line integrals (the Gradient theorem). You can define flux (the amount of stuff passing through a surface) as a surface integral of a vector field. I believe flux integrals have an application in electromagnetism, but I'm not 100% sure. If you want, you can show how to evaluate these with the Gauss-Ostrogradsky theorem (the Divergence theorem).  All of this is with integral calculus.

On the differential calculus side, you can start defining tangent vectors, normal vectors, and binormal vectors to vector curves (curves in 3d space). From these you can derive the formulas for normal and tangent acceleration, if you wish. Two other useful properties of vector curves are curvature and torsion. Both of these uniquely define a vector curve up to rigid translation/rotation (this is called the fundamental theorem of curves). Also take a look at the Frenet-Serret equations, which show why this is the case. As an application, with curvature you can find the speed limit of car travelling along a path defined by a vector curve so that the car does not skid off. You can also find something called the osculating plane and the osculating circle, which give nice approximations to vector curves. 

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7 hours ago, TheNintendoChip said:

Depends on how much you want to learn or what you're interested in. I just took the course so if you need any help with anything, let me know.

Vector functions relate a lot to physics. For example, you can use a vector function to parametrize the trajectory of a particle (notably when the path is parabolic). You can use a line integral over a vector field to find the force done by a particle traversing the field. You can then go into conservative vector fields and the fundamental theorem for line integrals (the Gradient theorem). You can define flux (the amount of stuff passing through a surface) as a surface integral of a vector field. I believe flux integrals have an application in electromagnetism, but I'm not 100% sure. If you want, you can show how to evaluate these with the Gauss-Ostrogradsky theorem (the Divergence theorem).  All of this is with integral calculus.

On the differential calculus side, you can start defining tangent vectors, normal vectors, and binormal vectors to vector curves (curves in 3d space). From these you can derive the formulas for normal and tangent acceleration, if you wish. Two other useful properties of vector curves are curvature and torsion. Both of these uniquely define a vector curve up to rigid translation/rotation (this is called the fundamental theorem of curves). Also take a look at the Frenet-Serret equations, which show why this is the case. As an application, with curvature you can find the speed limit of car travelling along a path defined by a vector curve so that the car does not skid off. You can also find something called the osculating plane and the osculating circle, which give nice approximations to vector curves. 

 

Hey, Thanks so much, but I feel like this is going a bit too much towards Physics, and not towards math, with that in mind, would you suggest I look more into the differential calculus side of vector functions?

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11 hours ago, chumbuks said:

Hey, Thanks so much, but I feel like this is going a bit too much towards Physics, and not towards math, with that in mind, would you suggest I look more into the differential calculus side of vector functions?

I mean, all the physics stuff is based on the math theory behind it. It's just a possible application, if that's what interested you. I like math theory, so the interesting part for me was proving all the theorems above. It helps to contextualize what you're doing. But I did my IA solely on an approximation method for a function that I created, which is all theory.

I personally prefer the differential calculus part. It's a lot easier to grasp, especially if you have no experience with multivariable calculus. That should work out better for your IA, because you don't want to have a bunch of technical stuff that you can't explain [Criterion E]. 

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