moonkey said:Homework Statement
Let C be a simple closed plane curve in space. Let n = ai+bj+ck be a unit vector normal to the plane of C and let the direction on C match that of n. Prove that
(1/2)∫[(bz-cy)dx+(cx-az)dy+(ay-bx)dz]
equals the plane area enclosed by C.
What does the integral reduce to when C is in the xy-plane?
Homework Equations
Stoke's Theorem
∫F.ds=∫(∇×F).dS
The Attempt at a Solution
I really have no idea where to start. Any help would be much appreciated.
LCKurtz said:Why don't you start by calculating ##\nabla \times \vec F## and plug it in the right side of Stokes' Theorem?
LCKurtz said:You are given a line integral, written in the form ##\oint \vec F\cdot d\vec R##. Can't you pick ##\vec F## out of that?
Stoke's Theorem is a fundamental theorem in vector calculus that relates the surface integral of a vector field over a closed surface to the line integral of the same vector field along the boundary of the surface. It is important because it provides a powerful tool for calculating integrals in three-dimensional space, as well as for understanding the relationships between different types of integrals.
A plane curve is a continuous curve in two-dimensional space that lies on a single plane. In the context of Stoke's Theorem, a plane curve is used to define a boundary for the surface over which the integral is being calculated. The curve is typically defined by a parametric equation, which allows for the calculation of the line integral using the curve's tangent vector.
To prove Stoke's Theorem for a plane curve, we first have to show that the line integral along the curve can be written as a double integral over the region enclosed by the curve. This is done by dividing the curve into small line segments and using the definition of a line integral to approximate the integral. Then, we use the definition of a surface integral to show that the double integral can be written as an integral over the surface enclosed by the curve. Finally, we compare this surface integral to the original line integral to prove Stoke's Theorem.
Yes, Stoke's Theorem is only applicable to a plane curve if the curve is smooth, meaning that it has a continuous tangent vector and no sharp corners. Additionally, the curve must be closed, meaning that it forms a complete loop and has no endpoints. These conditions ensure that the integrals involved in Stoke's Theorem are well-defined and can be calculated accurately.
Stoke's Theorem can be used for any type of surface, not just for plane curves. However, the theorem is most commonly used for surfaces that can be parameterized by two variables, such as a plane or a cylinder. For more complex surfaces, Stoke's Theorem may need to be modified or a different theorem may be more applicable.