Vector Integration: Line Integral

In mathematics, a line integral (sometimes called a path integral, contour integral, or curve integral; not to be confused with calculating arc length using integration) is an integral where the function to be integrated is evaluated along a curve.

The function to be integrated may be a scalar field or a vector field. The value of the line integral is the sum of values of the field at all points on the curve, weighted by some scalar function on the curve (commonly arc length or, for a vector field, the scalar product of the vector field with a differential vector in the curve). This weighing distinguishes the line integral from simpler integrals defined on intervals. Many simple formulae in physics (for example,W=F·s) have natural continuous analogs in terms of line integrals (W=∫_C F· ds). The line integral finds the work done on an object moving through an electric or gravitational field, for example.

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Equilibrium of Concurrent and Coplanar Force System

The conditions of equilibrium for concurrent force systems are obtained by determining the equations that produce a zero resultant. The magnitude of the resultant of a concurrent force system is found by means of the equation:

Obviously, the resultant will be zero and equilibrium will exist when the following equations are satisfied:

These equations are known as the conditions of equilibrium. It is important to note that with two conditions of equilibrium, only two unknown quantities can be determined to create equilibrium of a concurrent force system. (from Engineering Mechanics, 2nd edition by Ferdinand Singer)

The presentation file below gives a thorough discussion on equilibrium of concurrent and coplanar force systems:

Superposition

Superposition is a good method to apply in analyzing circuits with multiple sources. The basic concept of this method is to analyze a circuit one source at a time. Using the Superposition Theorem, we remove all the independent sources, except one, and analyze that circuit for that one. Then we repeat the procedure for another source, and so on. Finally, the net result is found by summing all the single -source results. 

'Removing' Power Sources

If we have two voltage sources and we were to remove one source, and still have a complete circuit, we must replace the voltage source with a short circuit.

Source Transformation

In order to solve difficult and complicated circuits, it is important to simplify the given circuit for easier analysis. One way to simplify circuits is by Source Transformation. It is the process of using Ohm's Law to take an existing voltage source in series with a resistance, and replace it with a current source in parallel with the same resistance or vice-versa. This method can be applied to Thevenin's theorem and Norton's theorem.

Source Transformation for Independent Sources