Khan Academy: Calculus: Example of Closed Line Integral of Conservative Field

Example of taking a closed line integral of a conservative field

Video lesson introducing conservative vector fields and the multivariable chain rule. Shows that a vector field is conservative if the line integral is the same for the same starting and ending point regardless of the path taken. [17:14]

Video lesson first reviews the definite integral and then introduces the line integral. Shows how to visualize a line integral and set it up. [18:48]

Video lesson solving a closed line integral of a vector field by using Green's theorem. [10:31]

Video lesson solving a line integral of a vector field by showing that the vector field is conservative and using the idea of path independence to solve the integral. [10:18]

Video finishing the derivation of Green's theorem by showing that the closed line integral in a vector field with vectors only in the y-direction is equal to the double integral over the region contained within the closed curve of the...

Video starting to derive Green's theorem by showing that the closed line integral in a vector field with vectors only in the x-direction is equal to the negative double integral over the region contained within the closed curve of the...

Video showing how to find the closed line integral of the dot product of a vector field and a unit normal vector to figure out how fast particles are exiting the region formed by the closed curve. Green's theorem is applied and the...

Using definite integral to solve a word problem about the growth in the population of a town.

Video extending the idea of a conservative vector field to show that a closed curve line integral will be equal to zero by showing that the difference of a line integral of a path and a line integral of a path with the same end points...