Fundamental Theorem Calculus

In this limit, the last two integrals in Eq. (4-168) become 0. Then we can apply the fundamental theorem of calculus to get... 

Assuming that the required limit exists and that it can be calculated, the fundamental theorem of the integral calculus can be stated as follows. 

The boundary surface of a region in space is an important physical quantity. The integral of a field in the region is related by the fundamental theorem of calculus to an integral over its boundary surface. A surface integral can be approximated by summing quantities associated with a subdivision of the surface into patches. In the present work, the surface patches are taken to be the (approximate) exposed surface area of atom in a molecule. 

The fundamental theorem of calculus allows us to rewrite the left hand side as y(x +1) — y(xi). Thus for each i we obtain... 

This appendix briefly describes the fundamental theorems of tensor calculus, which are widely used in our book. The notation closely follows the monograph of Zhdanov... 

The fundamental theorem of calculus Given a continuous function/(x) on the bounded interval [0,x], then... 

As with derivatives, these limit calculations can be quite tedious. Methods have been discovered and proven that allow the limit process to be bypassed. The crowning achievement of the development of calculus is its fundamental theorem The derivative of a definite integral with respect to its upper limit is the integrand evaluated at the upper limit the value of a definite integral is the difference between the values of an antiderivative evaluated at the limits. If one is looking for the definite integral of a continuous f x) between x = a and x= b, one need only find any antiderivative f(x) and calculate T(Zi) -F a). 

The left-hand side of this equation contains a conventional symbol for a definite integral a vertical line segment following the antiderivative function with the lower limit written at the bottom and the upper limit written at the top. Equation (7.14) is often called tht fundamental theorem of integral calculus. The antiderivative function F is called the indefinite integral of the integrand function /. 

Whenever one of these functions goes through zero, the other has a local maximum at -h 1 or minimum at — 1. This follows easily from differential calculus, as we will show later. Pythagoras theorem translates to the fundamental trigonometric identity ... 

---