# Theory question about Exact Eqtns and Calculus

• flyingpig
In summary, there is a relationship between exact equations and vector field dot form, where if the differential satisfies certain conditions, there exists a scalar function and a line integral that is independent of path. This can help in solving exact equation problems by writing an arrow on top of the function.
flyingpig

## Homework Statement

I've noticed that exact equations take the same form as vector field dot <dx,dy> form, are there any relationships? Because I ended up writing an arrow on top of my F(x,y) everytime I do these exact equation problems

Is there a secret my book is not telling me?

not sure if i understand the question, but if you have a differential
$$dQ = A(x,y)dx + B(x,y)dy$$

and they satisfy
$$\frac{A(x,y)}{dy} = \frac{B(x,y)}{dx}$$

then there exists a scalar function Q(x,y), such that dQ
$$dQ = A(x,y)dx + B(x,y)dy$$

similarly if you consider the vector field
$$F = \begin{pmatrix} A(x,y) \\ B(x,y)\end{pmatrix}$$
a line integral will independent of path, depending only on endpoints and equal to $Q(\vec{x}_f) - Q(\vec{x}_i)$, and you could write:
$$dQ = F \bullet \vec{dx}$$

## 1. What is an exact equation?

An exact equation is a mathematical equation that can be solved exactly, meaning that there is a specific solution that satisfies the equation. This is in contrast to an approximate equation, where the solution is an estimate or approximation.

## 2. How is calculus used in solving exact equations?

Calculus is used in solving exact equations because it provides the tools and methods for finding the exact solution. This includes techniques such as differentiation and integration, which are essential for solving equations involving derivatives and integrals.

## 3. Can all equations be solved exactly using calculus?

No, not all equations can be solved exactly using calculus. There are certain types of equations, such as transcendental equations, that cannot be solved using traditional calculus methods. In these cases, approximate methods may be used to find a solution.

## 4. What is the difference between an exact equation and an approximate equation?

The main difference between an exact equation and an approximate equation is that an exact equation has a specific, exact solution that satisfies the equation, while an approximate equation only provides an estimate or approximation of the solution. Exact equations are typically seen in pure mathematics, while approximate equations are used more frequently in applied fields.

## 5. Why are exact equations important in science and engineering?

Exact equations are important in science and engineering because they allow for precise and accurate solutions to be found, which are necessary for making accurate predictions and calculations. In fields such as physics and engineering, even small errors in calculations can have significant consequences, making the use of exact equations essential.

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