Vector Calculus: Integral Theorems

In summary, the conversation revolves around solving question 3 part b and c using Divergence and Stokes Theorems and knowledge of parametrization. The person has already found the B field using curl, but is struggling with calculating the flux through the top hemisphere or the entire sphere. They have tried using divergence theorem, but it resulted in a flux of 0. They were told to use the right theorem and that ds will produce an easy component to calculate. They have also tried a line integral, but it resulted in a complicated answer. The person suggests reprogramming the latex reference to make it easier to use.
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Homework Statement



Question 3 part b and c


Homework Equations



Divergence and Stokes Theorems. Knowledge of parametrization ect ect



The Attempt at a Solution



I got the B field by using curl. However any attempt to resolve the flux through the top hemisphere or even the sphere as a whole just gives me a horrible mess filled with cos's, sines and exponentials.

I used divergence theorem and calculated that the flux = 0. This cannot be write otherwise there would be no need for part c. Stokes theorem is the one that gives me the horrible mess.

My lecturer says that I have to use the right theorem, and ds will produce 1 easy component to calculate. I've tried everything but literally it's impossible.

Can someone shine the light on what theorem to use?

I tried to just do a line integral since the sphere is bounded by x^2 + y^2 = 1... Again I get a horrible mess that I cannot integrate.

I would show my proper attempt but your latex reference is too longwinded and difficult to use. It would be better if you reprogrammed it to work like the equation editor on Microsoft Word.
 

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  • #2
Don't worry I got it.
 

1. What is the difference between a line integral and a surface integral?

A line integral is calculated along a curve or path, while a surface integral is calculated over a surface. Line integrals are typically used to calculate work or flux along a path, while surface integrals are used to calculate area or volume under a surface.

2. How do the Divergence and Stoke's Theorems relate to each other?

The Divergence Theorem is a special case of Stoke's Theorem, where the flux of a vector field through a closed surface is equal to the integral of the divergence of the field over the enclosed volume. In other words, the Divergence Theorem relates a volume integral to a surface integral.

3. Can the Fundamental Theorem of Calculus be applied to vector functions?

Yes, the Fundamental Theorem of Calculus can be extended to vector functions in the form of the Fundamental Theorem of Line Integrals. This theorem states that the line integral of a vector field along a curve can be calculated by evaluating the antiderivative of the vector field at the endpoints of the curve.

4. What is the significance of Green's Theorem?

Green's Theorem relates a line integral around a closed curve to a double integral over the region enclosed by the curve. It allows for the evaluation of a line integral using techniques from multivariable calculus, making it a useful tool in solving problems involving vector fields and curves.

5. How is vector calculus used in real-world applications?

Vector calculus is used in a variety of fields, including physics, engineering, and economics, to model and analyze complex physical systems. It is used to calculate quantities such as work, flux, and circulation, and has applications in fluid dynamics, electromagnetism, and optimization problems.

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