Seven Fundamental Theorems of Calculus Examples

The fundamental premise of calculus, set out in this article, is that a solution exists for every problem at the solution level. Any set of real numbers h and k can be used to show the solutions of a system of linear equations, i.e., a polynomial equation with parameters h, k, x, and y. Let us call this set of equations a model of the real world. The solutions of a model are also called the variables of that model. We shall therefore leave the description of solutions to the students, who will decide which of the models satisfy their queries.

Problems in geometry satisfy the following fundamental Theorems. First, for every point on the surface of a sphere, there exists a corresponding point on the curve connecting every two points on the surface. This follows by assuming that if a point on the curve falls off from a constant angle to the angle of inclination, then the corresponding point on the surface will also fall off to the same angle of inclination. Thus, if a point p is on the horizontal meridian inclined to the x axis and p is on the vertical meridian inclined to the y axis then there exists such a curve on the horizontal and vertical planes, namely a tangent to the function f(x, y, z). Therefore, if such a tangent exists then the slope of the tangent on the x-axis is equal to the slope of the tangent on the y-axis, or simply f(x, y, z).

The second theorem shows that for any function f(x, y, z), there exists distinct curve forms on the surfaces of spheres. These forms are called tangential and radial functions. For every point on the sphere a solution can be derived by locating such a point on the tangent curve to the function f(x, y, z), then taking the difference between the points on the curve. Thus if a point on the tangent plane is fixed then a solution exists for any function f(x, y, z) on the flat surface. But, if a point on the flat surface moves away from the fixed point then an analogous curve on the tangent plane can be obtained by locating the point on the tangent curve to the function f(x, y, z) on the curved surface.

The third Theorema demonstrates that the set of all functions on surfaces which satisfies the equations f(x, y, z) and g(x, y, z) for some set up is called a set of closed forms. These are called the embedding operators. The set x of embedding operators is equivalent to the set of all functions on surfaces which satisfies the equations f(x, y, z) and g(x, y, z) for some set xp. The set up of a function on the surface can be called the embedding operator. In other words, the set of all functions on surfaces is called the embedding operator.

The fourth Theorema demonstrates that the set of all functions on a plane which satisfies the equations f(x, y, z) and g(x, y, z) for some set up is called a function neighborhood. The set is of the embedding operator is equal to the set of all functions on the domain of that operator. Thus a set x of operator i such as coordinate changes is called an embedding operator. A function is called a neighborhood when it satisfies certain fixed points x and y such that for every k such that for every I such that for every k if k is true then the function f(x, y, z) is true. The set is of the operator i such as coordinate changes is called the set of embeddings of k on the domain of operator i. Thus, for every k, if k is true then the function f(x, y, z) is true and for every I such that for every k if k is true then the function f(x, y, z) is false.

The fifth Fundamental Theorema demonstrates the use of integral operators on functions of a single variable. The set of Integral operators is the set of all possible functions of a single variable. Therefore, for every k such that for every I such that for every k, if k is true then the function f(x, y, z) is true. These are called integral operators. In addition they are used to evaluate complex values in a similar way to the function on the real number line evaluated by a real number.

The sixth Fundamental theorem is to show that the set of all functions is not a closed one. In other words, if there is a sequence of Integrals then the sequence will continue forever. In other words all functions can be obtained by adding the Integrals.

The seventh Fundamental Theorema demonstrates the usefulness of integration. Let us consider an example using the integral of a set of real part. We will start by defining the set of real parts. Then we define the set of real x’s by means of the integral operator. Finally we will prove that each of these sets are also a unique set. This completes the first part of the seven Fundamental Theorems of calculus.