Theorems on vector calculus.

In summary, Green's theorem states that line integrals and surface integrals can be interchanged, and this is useful for calculations. Curl is a vector quantity that is important for understanding the dynamics of a field. It is defined at a single point and is not visible in the delta formulation of curl.
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I realize by now I must be making everyone crazy on this forum with my questions on vector calculus...but I really have no choice!
Please bear with me for another few days...I promise I'll get this done as fast and as painlessly as possible.

Here goes...

1.In divergence theorem,green's theorem and stoke's theorem,we basically interchange between types of integrals-e.g green's theorem let's us interconvert line integrals and surface integrals...do we need to interchange between types of integrals just to aid calculations?

2.Why does curl need to be a vector quantity? The concept curl(i.e the amount of rotation at a particular point) could be explained by a scalar quantity also...we wouldn't need to assign a direction?

3. Why is an irrotational field nonconservative?
(I found a website describing an irrotational field as one in which there is 'free energy'...what's this all about?)
 
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As per your first question, yes it does aid in calculations. The integrals you originally use can be done perfectly well, but it is typically far more convenient to do the simpler integral. It could also make identifications easier.. not that I have any examples off the top of my head.

For your second question, you DO want to know in what direction something curls though. It's like assigning a velocity to a scalar. You DO want to know what direction the certain velocity is in, thus we use vectors.
 
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I like your example about assigning a direction to speed.
Thanks.
 
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I promise I don't have many more questions to ask.

But here's another few..

1.The direction of curl is (as a convention ) along the direction prescribed by the right hand thumb rule...now,inside a closed curve,when we sum the curl vector (which is basically the limit of path integral around a point,per unit area around the point; as area tends to zero)the sum of all these path integrals is supposed to be equal to the path integral all around the actual (macroscopic) closed curve(by green's theorem).

My question is: by summing the curls at each point,we are summing the perpendicular vectors representing the curls...but conceptually,how can the sum of these perpendicular vectors represent the macroscopic path integral?

2. Why is curl defined at a single point(as I defined in the previous question)...wouldn't it have been more useful if curl had been the macroscopic circulation instead of the microscopic circulation(as in the total path integral around the entire closed curve)?

3. Lastly,curl may also be defined as the measure of change of a vector field in a direction perpendicular to it...but this definition is not 'visible' in the usual delta formulation of it...please explain if I'm wrong somewhere.
 

1. What is a vector in vector calculus?

A vector in vector calculus is a mathematical object that has both magnitude (size) and direction. It can be represented by an arrow, with the length of the arrow representing the magnitude and the direction of the arrow representing the direction.

2. What are the basic operations used in vector calculus?

The basic operations used in vector calculus are addition, subtraction, scalar multiplication, and dot and cross products. Addition and subtraction are used to combine vectors, scalar multiplication is used to scale a vector, and dot and cross products are used to find the angle between two vectors and the magnitude of their resulting vector, respectively.

3. What is the difference between a scalar and a vector in vector calculus?

A scalar in vector calculus is a mathematical quantity that has only magnitude and no direction. Examples of scalars include speed, temperature, and mass. A vector, on the other hand, has both magnitude and direction, as described in the first question. Examples of vectors include velocity, force, and displacement.

4. How are vectors represented in vector calculus?

Vectors are typically represented using either Cartesian coordinates (x, y, z) or polar coordinates (r, θ). In Cartesian coordinates, a vector is represented by its components along each axis, while in polar coordinates, a vector is represented by its magnitude and direction angle.

5. What are some real-world applications of vector calculus?

Vector calculus has many applications in the fields of physics, engineering, and computer graphics. It is used to model and analyze the motion of objects, such as projectiles and planets, and to calculate forces and work in mechanical systems. It is also used in computer graphics to create 3D animations and simulations.

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