Text says scalar is invariant in rotational transformation.Then to prove some functions are scalar field, should I do rotational transformation, or are there any other methods?
I'm so sorry. It was my first time to write in this forum. I know that cross product makes perpendicular vectors. But in this problem, I don't understand how we explain three dimension by using two parameter, u and v. I searched in internet and thought it is related to gradient. Is it right?
Homework Statement
Homework Equations
The Attempt at a Solution
I can't understand what is ɛ in this problem, and why should we adopt it. Could you explain me please?
Homework Statement
Homework Equations
The Attempt at a Solution
I solved #2,4 but I don't understand what #1,3 need to me. I know that scalar field is a function of points associating scalar value. But how can I prove some function is scalar field or vector field?
Homework Statement
Let us consider three scalar fields u(x), v(x), and w(x).
Show that they have a relationship such that f(u, v, w) = 0 if and only if
(∇u) × (∇v) · (∇w) = 0.
Homework Equations
The Attempt at a Solution
I can do nothing but just writing components of (∇u) × (∇v) · (∇w)...
Homework Statement
Homework Equations
The Attempt at a Solution
I could find how to solve #2,4, but I don't understand what #1,3 need to me. How can I prove some functions are scalar field or vector field?
Homework Statement
Let us consider three scalar fields u(x), v(x), and w(x). Show that they have a relationship such that f(u, v, w) = 0 if and only if
(∇u) × (∇v) · (∇w) = 0.
Homework Equations
The Attempt at a Solution
I could think nothing...