Time Dependent Vector Field - Relationship With Vector Fields in The Usual Sense

Relationship With Vector Fields in The Usual Sense

A vector field in the usual sense can be thought of as a time dependent vector field defined on even though its value on a point does not depend on the component .

Conversely, given a time dependent vector field X defined on, we can associate to it a vector field in the usual sense on such that the autonomous differential equation associated to is essentially equivalent to the nonautonomous differential equation associated to X. It suffices to impose:

for each, where we identify with . We can also write it as:

.

To each integral curve of X, we can associate one integral curve of, and vice versa.

Read more about this topic:  Time Dependent Vector Field

Famous quotes containing the words relationship with, relationship, fields, usual and/or sense:

    Some [adolescent] girls are depressed because they have lost their warm, open relationship with their parents. They have loved and been loved by people whom they now must betray to fit into peer culture. Furthermore, they are discouraged by peers from expressing sadness at the loss of family relationships—even to say they are sad is to admit weakness and dependency.
    Mary Pipher (20th century)

    It is possible to make friends with our children—but probably not while they are children.... Friendship is a relationship of mutual dependence-interdependence. A family is a relationship in which some of the participants are dependent on others. It is the job of parents to provide for their children. It is not appropriate for adults to enter into parenthood recognizing they have made a decision to accept dependents and then try to pretend that their children are not dependent on them.
    Donald C. Medeiros (20th century)

    The greatest delight which the fields and woods minister, is the suggestion of an occult relation between man and the vegetable. I am not alone and unacknowledged. They nod to me, and I to them.
    Ralph Waldo Emerson (1803–1882)

    It is usual for a Man who loves Country Sports to preserve the Game in his own Grounds, and divert himself upon those that belong to his Neighbour.
    Joseph Addison (1672–1719)

    One great advantage which poetry has over prose—one sense in which, we might even say, it is considerably more beautiful—is that it fills up space approximately three times as rapidly.
    James Thurber (1894–1961)