Relation between parameters of a vector field and it's projection

In summary, when forming the projection of Y, Y' orthogonal to X, the parameters of Y and Y' can be related through the expression Y' = Y - (scalar multiple of X). However, the curves for Y and Y' may not always have a direct relationship, as the curve for Y' depends on the vector field X. In some cases, it is possible to solve for the curves to determine the relationship between the parameters.
  • #1
center o bass
560
2
Say we have two vector fields X and Y and we form the projection of Y, Y' orthogonal to X. Since every vector field is associated with a curve with a corresponding parameter, is there a relation between the parameters of Y and Y'?
 
Physics news on Phys.org
  • #2
The projection of Y orthogonal to X can be expressed as (a scalar multiple of)

$$Y' = Y - \frac{\langle X, Y\rangle X }{ \langle X, X \rangle} ,$$

wherever ## \langle X, X \rangle\neq 0##. We can define the flow curves via ##\dot{\gamma}_Y(t) = Y \gamma(t)## and ##\dot{\gamma}_{Y'}(t) = Y' \gamma(t)##. In principle we can use the same parameter ##t## to describe both of these curves, but the curve for ##Y'## will also depend on the vector field ##X##, so I don't think there's a general answer to your question. If ##X## and ##Y## are given and nice enough, then we can just solve for the curves to work out the relationship.
 

1. How do the parameters of a vector field affect its projection?

The parameters of a vector field, such as its direction and magnitude, determine the behavior of its projection. For example, a vector field with a higher magnitude will have a larger projection, while a vector field with a different direction will have a different projection. Additionally, the parameters of a vector field can also affect the shape and orientation of its projection.

2. Can the parameters of a vector field change the direction of its projection?

Yes, the direction of a vector field's projection is directly influenced by its parameters. If the parameters of the vector field change, the direction of its projection will also change accordingly. For example, if the magnitude of the vector field increases, the direction of its projection will also change to reflect this increase.

3. What is the relationship between the magnitude of a vector field and its projection?

The magnitude of a vector field directly affects the length of its projection. A vector field with a higher magnitude will have a longer projection, while a vector field with a lower magnitude will have a shorter projection. This relationship is crucial in understanding the behavior of vector fields and their projections.

4. How do the parameters of a vector field impact the shape of its projection?

The parameters of a vector field, such as its direction and magnitude, can greatly influence the shape of its projection. For instance, a vector field with a consistent direction will have a projection that is linear and uniform in shape. On the other hand, a vector field with varying direction can produce a more complex and irregular projection shape.

5. Is there a mathematical formula to calculate the projection of a vector field?

Yes, there is a mathematical formula to calculate the projection of a vector field. It involves using the dot product between the vector field and the unit vector in the direction of the projection. This formula can be applied to any vector field, regardless of its parameters, to determine its projection accurately.

Similar threads

  • Differential Geometry
Replies
20
Views
2K
  • Differential Geometry
Replies
2
Views
1K
Replies
4
Views
1K
  • Differential Geometry
Replies
1
Views
1K
  • Differential Geometry
Replies
2
Views
1K
  • Differential Geometry
Replies
1
Views
2K
  • Differential Geometry
Replies
16
Views
2K
Replies
8
Views
1K
  • Differential Geometry
Replies
7
Views
2K
  • Differential Geometry
Replies
12
Views
3K
Back
Top