  In this paper we study the relationship of the Brouwer degree of a vector
field with the dynamics of the induced flow. Analogous relations are studied
for the index of a vector field. We obtain new forms of the Poincar% \'{e}-Hopf
theorem and of the Borsuk and Hirsch antipodal theorems. As an application, we
calculate the Brouwer degree of the vector field of the Lorenz equations in
isolating blocks of the Lorenz strange set.
