Fundamental Theorem for Line Integrals

In summary, the conversation discusses a vector field F(bar) and its components, fx(x,y) and fy(x,y). It also introduces a function f(x,y) and its derivative fy(x,y), along with a specific curve C. The conversation ends with finding the integral of F(bar) along C and using the previously discussed elements to find the solution.
  • #1
andyk23
26
0
Vector field F(bar)= <6x+2y,2x+5y>
fx(x,y)= 6x+2y fy(x,y)= 2x+5y
f(x,y)= 3x^2+2xy+g(y)
fy(x,y)=2x+g'(y)
2x+g'(y)= 2x+5y
g'(y)= 5y
g(y)= 5/2*y^2
f(x,y)=3x^2+2xy+(5/2)y^2
Then find the [itex]\int[/itex] F(bar)*dr(bar) along curve C t^2i+t^3j, 0<t<1
I'm stuck on finding the last part for the F(bar) would I use <6x+2y,2x+5y> and substitute the <t^2,t^3> in the for x&y then do F(bar)* rbar'(t)
so [itex]\int[/itex] from 0 to 1 of <6t^2+2t^3,2t^2+5t^3> <2t,3t^2>
Thanks for the help
 
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  • #2
You figured out a potential for your vector field. You could use that to do the integral. And then you set up the explicit contour integral. Both look correct. They give you the same answer, don't they?
 

FAQ: Fundamental Theorem for Line Integrals

What is the Fundamental Theorem for Line Integrals?

The Fundamental Theorem for Line Integrals is a mathematical theorem that relates the value of a line integral along a closed curve to the values of a function at the endpoints of the curve. It states that if a function has a continuous gradient over a region, the line integral of that function over a closed curve in that region can be calculated by evaluating the function at the endpoints of the curve and taking the difference.

What is the significance of the Fundamental Theorem for Line Integrals?

The Fundamental Theorem for Line Integrals is significant because it allows for the calculation of line integrals in a more efficient and straightforward manner. It also provides a connection between line integrals and gradient fields, which are important concepts in vector calculus and physics.

What is the difference between a line integral and a normal integral?

A line integral is a type of integral that calculates the total value of a function along a specific path or curve, whereas a normal integral calculates the area under a curve. Line integrals are used to determine quantities such as work, energy, and fluid flow along a specific path.

What are the conditions for the Fundamental Theorem for Line Integrals to hold?

The Fundamental Theorem for Line Integrals holds when the function being integrated has a continuous gradient over the region of integration. This means that the function must be differentiable and have a continuous derivative at all points within the region.

How is the Fundamental Theorem for Line Integrals related to Green's Theorem?

The Fundamental Theorem for Line Integrals and Green's Theorem are closely related, as they both involve the evaluation of line integrals. Green's Theorem is a special case of the Fundamental Theorem for Line Integrals, and it can be used to evaluate line integrals over simple closed curves in the plane.

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