how vector area of a closed surface is zero?
You should explain the context, as well as your mathematical experience. There are varying ways we can answer your question. The best way to answer your question, if I'm understanding it correctly, goes beyond multivariate calculus. What course is this for?Manisha Punia said:how vector area of a closed surface is zero?
I did say the best way, right? The best way, in my opinion, is via the generalized Stokes' Theorem.lurflurf said:Mandelbroth can you explain what you mean by goes goes beyond calculus? That is very much a first year calculus question.
Not the Stokes' Thoerem I'm talking about. :tongue:lurflurf said:Stokes' Theorem is a fundamental theorem of calculus. Some might say Stokes' Theorem is the fundamental theorem of calculus. Granted as I said above it involves other mathematics (ie algebra and geometry) and is difficult to show in general or for difficult specific examples. Many calculus books have Stokes' theorem towards the end and/or do little with it. A big problem is that calculus is a hodge podge of random techniques and not a unified subject.
A closed surface is a three-dimensional shape that completely encloses a finite amount of space. It has no boundary and can be represented by a continuous vector field.
The vector area of a closed surface is calculated by integrating the cross product of the surface's unit normal vector and the differential area vector over the entire surface.
The vector area of a closed surface is always zero because the integral of the cross product of the unit normal vector and the differential area vector is equal to zero. This is due to the fact that the surface has no boundary, so the contributions from each point on the surface cancel each other out.
No, the vector area of a closed surface can never be non-zero. This is a fundamental property of closed surfaces in vector calculus.
The vector area of a closed surface being zero means that there is no net flux of the vector field across the surface. This has important applications in physics, such as in the study of electric and magnetic fields.