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Jhenrique
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Is possible to compute indefinite integrals of functions wrt its variables, but is possible to compute indefinite integrals of scalar fields and vector fields wrt line, area, surface and volume?
Matterwave said:You tell me. When do you think a path integral makes any sense if you don't specify a path? When do you think a surface integral makes any sense if you don't specify a surface?
?$$\\ \int \vec{f} \cdot d\vec{r} = \int \vec{\nabla}\phi\cdot d\vec{r}=\phi$$
Jhenrique said:Yeah, but I'm talking about a indefinite integral of a vector field! This can exist? This make sense?
Matterwave said:If you are unable to even think critically about a question, it is best to learn a bit more about the subject first before making conjectures. To be honest Jhenrique, not to discourage you or your love of science/math, but a lot of your conjectures seem to make no sense. Perhaps they can lead somewhere if you can at least formulate the questions in a cogent manner.
There's a big book by Arfken and Weber called Mathematical Methods for Physicists (I'm a physicist, so I learn from physics textbooks) that covers basically everything you're worried about. You can use it as a study guide or a reference text. It's quite comprehensive. However, it looks at things from a physicist's perspective, so it's not always 100% rigorous like a math text would be. You might also want to pick up a pure math book on the subject and see where that takes you as well.
Jhenrique said:When is about vector and tensor calculus my doubts are exponentially big... :(
More one thing... indefinite integrals requires (implicitly or not) path independence? In other words, only is possible to compute indefinite integrals of exact form?
An indefinite integral of fields is a mathematical concept used in physics to represent the total amount of energy or other physical quantity in a given region of space. It is calculated by finding the antiderivative of a given function, which describes the behavior of the field at each point in space.
The main difference between an indefinite and a definite integral is the inclusion of limits of integration. In a definite integral, the limits are specified and the resulting value is a single number. In an indefinite integral, the limits are not specified, and the resulting value is a function that describes the entire field.
An indefinite integral of fields represents the total amount of a physical quantity (such as energy or electric charge) in a given region of space. This can be used to calculate the behavior of the field at any point within that region, and can also be used to calculate the flow of the field between different regions.
Indefinite integrals of fields have numerous applications in physics and engineering. They are used to calculate the potential energy in a gravitational or electric field, the work done by a force, and the charge or current flowing through a circuit. They are also used in the study of fluid mechanics, electromagnetism, and other fields of physics.
Some common techniques for evaluating indefinite integrals of fields include substitution, integration by parts, and partial fractions. These techniques involve manipulating the given function to make it easier to integrate, and then using standard integration rules to find the antiderivative. In some cases, it may also be necessary to use numerical methods or computer software to evaluate the integral.