Line Integral Fundamental Theorem

In summary, Using the given Phi and the fundamental theorem of line integrals, the line integral from (0,2) to (-2,0) is evaluated to be 4, which may not be the same answer as before due to a possible error in calculation.
  • #1
killersanta
63
0

Homework Statement


Use Your Phi(from part 1) and the fundamental theorem of line integrals to evaluate the same line integral. (should get the same answer!)



The Attempt at a Solution



Phi from part 1: Phi = xy+ y^2 +C

The line from before go from (0,2) to (-2,0)

r(a) = (0,2) r(b)= (-2,0)

Gradient F * Dr = f(r(b))-f(r(a))
= f(-2,0)-f(0,2)
= (-2*0+0^2)- (0*2+2^2)=4

I'm not sure if this is right, it's not the same answer I got before, I'm not sure what one I got wrong.
 
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  • #2
For one, you're clearly missing a minus sign.
 
  • #3
snipez90 said:
For one, you're clearly missing a minus sign.

Other than that is it good?
 

What is the Line Integral Fundamental Theorem?

The Line Integral Fundamental Theorem is a fundamental theorem in multivariable calculus that relates the line integral of a vector field to the values of a scalar potential function at the endpoints of the curve over which the integral is calculated.

What is the difference between the Line Integral Fundamental Theorem and the Fundamental Theorem of Calculus?

The Fundamental Theorem of Calculus relates the integral of a function to its antiderivative, while the Line Integral Fundamental Theorem relates the line integral of a vector field to a scalar potential function. Additionally, the Fundamental Theorem of Calculus is for single-variable calculus, while the Line Integral Fundamental Theorem is for multivariable calculus.

What is the significance of the Line Integral Fundamental Theorem in real-world applications?

The Line Integral Fundamental Theorem is used in fields such as physics, engineering, and economics to calculate work done by a force, flow of a fluid, and work done by a financial force, respectively. It is also used in vector calculus to solve problems related to conservative vector fields and path independence.

How is the Line Integral Fundamental Theorem derived?

The Line Integral Fundamental Theorem is derived using the gradient theorem and Green's theorem, which relate the line integral over a closed curve to the double integral over the region enclosed by the curve. By equating these two integrals and using the properties of conservative vector fields, the Line Integral Fundamental Theorem can be derived.

Are there any limitations to the Line Integral Fundamental Theorem?

While the Line Integral Fundamental Theorem is a powerful tool in vector calculus, it does have some limitations. It can only be applied to conservative vector fields, and the scalar potential function must be continuous and differentiable over the region of integration. Additionally, it only applies to line integrals over closed curves.

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