Synopsis. Linear algebra is a fairly extensive subject that covers vectors and matrices, determinants, systems of linear equations, vector spaces and linear transformations, eigenvalue problems, and other topics. As an area of study it has a broad appeal in that it has many applications in engineering, physics, geometry, computer science, economics, and other areas. It also contributes to a deeper understanding of mathematics itself.
Synopsis. Vector fields, as geometric objects and/or functions, provide
a backbone in which all of physics and engineering, really mathematical
modeling is structured on. From force fields in physics to slope fields in
differential equations and modeling, the notion of a vector field allows us
to recover measurable quantities from models defined only by equations of
motion. Here, we begin the study of their basic structure and properties.
Synopsis. Today, we go directly into on of the three big theorem's of
vector calculus, Green's Theorem. This theorem exposes a deep relationship
between the aggregate behavior of a vector field along the boundary of a
relatively nice region in the plane (the vector line integral), to the integral
of a related function on the interior of the region. Since Green's Theorem
is restricted to regions in the plane, there are a number of ways to craft
the related integrals, giving different geometric meaning to the quantities.
One interesting geometric interpretation is that the theorem relates the total
twisting effect of a vector field in the region (measured by integrating the
curl of the vector field as it sits in three space with no vertical component), to
the total tangent component of the vector field along the closed boundary.
Synopsis. In this lecture, we begin to finish the foundational material
of what makes a vector calculus course with a full discussion of one of the
two other Big Theorems, those of Stokes and Gauss. Here, we present
and discuss Stokes' Theorem, developing the intuition of what the theorem
actually says, and establishing some main situations where the theorem is
relevant. Then we use Stokes' Theorem in a few examples and situations.
- Teacher: ABDELAZIZ MAOUCHE