Bolza’s Fundamental Theorem History Project

Posted: August 26th, 2021

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Bolza’s Fundamental Theorem History Project

Bolza formulated Bolza’s Fundamental Theorem in 1913 (Goldstine 48). It is a classical problem in mathematics for solving calculus of variations using differential equations. It links the concept of differentiating a function with integrating a function. The theorem can briefly describe a broad-spectrum problem in calculus that has maxima and minima comparable to more straightforward problems in modern calculus of variations (Goldstine 56). Hence, the essay analyzes Bolza’s theorem, which is a continuous function on a closed bounded interval and has the first and second sections (Goldstine 56). Equally, the paper argues that the theory has two functions of calculus that are integration and differentiation.

Analysis and Interpretation

The problem statement in Bolza’s Fundamental theorem is as follows;

1. Maximize

Where:

The following equations should be satisfied;

Where:

Where:

Optimize the below equations;

Bolza’s theorem is applied in integration and differentiation. In this case, it has two versions in the calculus of variations as shown below;

1. First Version

The continuous functionis given at time  for the change in and. In this equation, the function  is expressed as a derivative of andis anti-derivative of. A new function can be articulated for the area below the curve as , a function of function at time t as shown below;

Thus, the theorem is used to calculate the area under the curve using anti-differentiation.

• Second Version

The equation offers a more explicit method of calculating the area below a curvebetween and by getting the anti-derivative of using

Discussion

It is a simple-looking model that directly correlates with hypotheses mildness used to obtain conclusive results. In the optimization equation, the endpoints and integrandshould be non-differentiable and can be infinite (Goldstine). The features in the model allowenormous flexibility in calculus, where endpoint constraints and the differential is introduced in  and. Therefore, the only direct constraint in the equation is an open set , closed set  at timebetweenand. The theorem has much applicability as it uses symbolic integration instead of numeric integration (Goldstine 213). It can be used in cost analysis, hydro-thermal systems, mechanics, and engineering, as well as linear programming.

Conclusion

The most significant drawback of the theorem is used in the application in non-linear programming as it brings about convex Bolza problems in the calculus of variations and optimal controls. Also, in areas where the period is fluctuating too much, it can bring difficulties in integration and differentiation as they are opposite of each other, leading to non-smooth data. Hence, non-convex and non-smooth data leads to duality results; optimum conditions are not obtained due to sub-differentials and transverse circumstances. Therefore, the model is excellent in calculating variations when the constraints are well stated with consistent data that is correctly interpreted.

Works Cited

Goldstine, H. H. A History of the Calculus of Variations from the 17th through the 19th Century. Springer Science & Business Media,2012.