I believe it is time we have an interpretation to the line integral. So far, we are have seen the conditions of the line integral and the technique we use to evaluate it, which is nothing more than finding the dot product, substituting the component functions for x, y, and z and integrating accordingly.
We now borrow some principles in physics for its interpretation.
VECTOR INTEGRAL CALCULUS >
INTERPRETATION OF THE LINE INTEGRAL
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LINE INTEGRAL OF A PIECEWISE SMOOTH CURVE
GREEN'S THEOREM
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VIDEO LECTURE
MAIN CONCEPTS
The line integral  gives us the work done by vector field  to move an object along the curve given by position vector  .Get the Printer Friendly Version COMMENTS
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CONCEPT
In our study of vector fields in vector calculus, recall that the vector
 is a vector field that gives a vector at any point in three-space. In physics, we say that this force is acting on an object in the space. It doesn’t require much intuition when I quote an example of say walking in water. Whatever point your body is in the water, there is a force that is acting on your body, the same force given by  .
When we introduce the concept of a curve, we say that this is a force moving an object along C from its initial point Let’s recall some physics. Suppose a constant force I drew the vector such that We know that the work done by this component of vector
Or in other words, the work done by Going back to our curve C, we will derive an expression for the work done as follows. Choose a point
The point will experience a force Therefore we sum
is the total work done by the vector field
which is the line integral. From this discussion, we see that the work done by the vector field will be different if the object moves along another curve. This changes the This emphasizes again that the line integral needs both a vector and a position vector for it to make sense. All information presentated, less questions and exercises, is original content of Donny, with slight references to various books.
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to its terminal point 
. Our main argument is that 
is acting on an object moving along vector 
as shown.
. Well, this can certainly be the case. Taking the walking in water example, I could be walking straight while the water is pushing me to the left. We are thus interested in the component of vector 
that is pushing the box along 
.
is given by 
to move the object along 
.
on the curve C.
. To find the work done by this force at this instant, we need to know the direction of travel of point 
at this same instant. Well, that isn’t too hard to find because the direction of travel is simply given by the tangent vector or simply 
.
over the entire curve by integrating giving us the conclusion that
. This is exactly the same as
