- #1
WMDhamnekar
MHB
- 376
- 28
- Homework Statement
- https://mathhelpforum.com/attachments/1667128375169-png.45062/
- Relevant Equations
- ##F=\langle z,x,y \rangle, z = 2x + 2y -1## and ##z = x^2 + y^2 ##
My answer:
I computed line integral as follows:vela said:You haven't used Stokes' theorem. You just calculated the first integral. (I assume. I didn't check your work.) Stokes' theorem tells you that if you were to calculate the other integrals, you will get the same result. So in this problem, you're really just verifying that the theorem works in this particular case, not actually using it.
So forget about using the theorem and just calculate the other integrals.
If my parametrization of the curve is wrong, what is your suggestion for correct parametrization of the curve?Orodruin said:Your parametrization does not describe the correct curve.
I believe it would be more instructive if you first verified that it is indeed wrong and then think about what you could do to correct it. What do you get if you insert the parametrization in, for example, the equation for the paraboloid ##z = x^2 + y^2## or the plane ##z = 2x + 2y - 1##?WMDhamnekar said:If my parametrization of the curve is wrong, what is your suggestion for correct parametrization of the curve?
A line integral is a mathematical concept used in vector calculus to calculate the work done by a force along a path. It involves integrating a vector function along a curve or line.
The purpose of a line integral is to calculate the work done by a force along a specific path. It is also used in many other applications such as calculating the flux of a vector field through a surface.
Stokes' theorem is a fundamental theorem in vector calculus that relates line integrals to surface integrals. It states that the line integral of a vector field over a closed curve is equal to the surface integral of the curl of that vector field over any surface bounded by the curve.
Stokes' theorem is used in many real-world applications, such as fluid dynamics, electromagnetism, and engineering. It allows for the calculation of surface integrals, which are important in determining quantities like fluid flow and electromagnetic flux.
One common misconception is that line integrals can only be calculated over straight lines. In reality, they can be calculated over any curve or path. Another misconception is that Stokes' theorem only applies to two-dimensional surfaces, when in fact it can also be applied to three-dimensional surfaces.