Relationship To The Divergence Theorem
Considering only two-dimensional vector fields, Green's theorem is equivalent to the following two-dimensional version of the divergence theorem:
where is the outward-pointing unit normal vector on the boundary.
To see this, consider the unit normal in the right side of the equation. Since in Green's theorem is a vector pointing tangential along the curve, and the curve C is the positively-oriented (i.e. counterclockwise) curve along the boundary, an outward normal would be a vector which points 90° to the right, which would be . The length of this vector is . So
Now let the components of . Then the right hand side becomes
which by Green's theorem becomes
The converse can also easily shown to be true.
Read more about this topic: Green's Theorem
Famous quotes containing the words relationship to the, relationship to, relationship and/or theorem:
“... the Wall became a magnet for citizens of every generation, class, race, and relationship to the war perhaps because it is the only great public monument that allows the anesthetized holes in the heart to fill with a truly national grief.”
—Adrienne Rich (b. 1929)
“Whatever may be our just grievances in the southern states, it is fitting that we acknowledge that, considering their poverty and past relationship to the Negro race, they have done remarkably well for the cause of education among us. That the whole South should commit itself to the principle that the colored people have a right to be educated is an immense acquisition to the cause of popular education.”
—Fannie Barrier Williams (1855–1944)
“The relationship between mother and professional has not been a partnership in which both work together on behalf of the child, in which the expert helps the mother achieve her own goals for her child. Instead, professionals often behave as if they alone are advocates for the child; as if they are the guardians of the child’s needs; as if the mother left to her own devices will surely damage the child and only the professional can rescue him.”
—Elaine Heffner (20th century)
“To insure the adoration of a theorem for any length of time, faith is not enough, a police force is needed as well.”
—Albert Camus (1913–1960)