Why not start by applying the definition of the divergence of a vector field to ##\phi\mathbf u##? Please don't use the empty set symbol instead of ##\phi##. I found it very confusing, and it took me some time to understand what you meant. If you don't use LaTeX, and can't find another way to type a ##\phi##, then just call the scalar field f or something like that.bothcats said:Homework Statement
div(øu) = ødivu + ugradø
Homework Equations
divergence of scalar field = f,ii
divergence of vector field = ui,i
The Attempt at a Solution
I've heard this is a simple proof, but this is my first one of 8 or so proofs I need to complete for homework, and I'm really not sure where to start. I know that div v = ∇ . v, but that's as far as I've gotten. We need to use Index Notation. Thoughts on where to start?
Vector calculus identities are a set of mathematical equations that relate various vector operations, such as differentiation and integration, to each other. These identities are used to simplify and manipulate vector equations in order to solve problems in physics, engineering, and other fields.
Some common vector calculus identities include the product rule, chain rule, gradient, divergence, and curl identities. These identities are used to express various properties of vector fields, such as their rates of change and directional derivatives.
Vector calculus identities are used in a wide range of real-world applications, including fluid dynamics, electromagnetism, and mechanics. They are used to model and analyze physical systems, and to solve problems related to the behavior and properties of vector fields.
Understanding vector calculus identities is essential for many fields of science and engineering, as they provide a powerful tool for solving complex problems involving vector quantities. They also provide a deeper understanding of the relationships between different mathematical operations and how they can be applied to real-world situations.
Improving your understanding of vector calculus identities requires practice and familiarity with the various identities and their applications. You can also benefit from studying examples of how these identities are used in different fields, and seeking guidance from textbooks, online resources, or a tutor.