- #1
connor415
- 24
- 0
u is a vector field,
show that
(u.∇)u = ∇(1/2u^2)+w∧u
Where w=∇∧u
show that
(u.∇)u = ∇(1/2u^2)+w∧u
Where w=∇∧u
connor415 said:ps I tried to do it starting from the left, could you do it that way please? Thanks
A vector field identity is a mathematical equation that relates two vector fields. It shows that the two fields are equivalent or have a similar structure.
(u.∇)u (read as "u dot gradient of u") is a notation used to represent the dot product between the vector field u and the gradient operator ∇. In this proof, it is used to show the relationship between the two vector fields.
∇(1/2u^2) (read as "gradient of one-half u squared") represents the gradient of the scalar function 1/2u^2. This term is often used in equations involving energy or potential, as it represents the change in energy or potential with respect to position.
w∧u (read as "w wedge u") represents the cross product between the vector fields w and u. In this proof, it is used to show the relationship between the two vector fields and indicate how they are related to each other.
This vector field identity is useful in many areas of science, including physics, engineering, and mathematics. It can be used to simplify and solve equations involving vector fields, as well as to analyze and understand the relationship between different vector fields. It also has applications in fluid dynamics, electromagnetism, and other fields that involve vector quantities.