Chapter 9 - Vector Calculus

Abstract

This chapter provides an overview of vector calculus, where a vector is a representation of a quantity and a direction. The dot product is also called the scalar product since it turns two vectors into a scalar quantity. Another lesser-used name for the dot product is the inner product. The dot product of a vector with itself produces the magnitude, squared of that vector. When one row is a linear combination of another row, the determinant is equal to zero. Two parallel vectors are linear combinations of each other. Hence, the triple scalar product is zero if any two vectors are parallel. When two rows are interchanged, the sign on the determinant changes. The surface integral is useful in many calculations ranging from how much sound comes from a surface to strength of attraction between two magnets. The surface integral is moving the standard integral into two dimensions. The energy required to overcome the friction along the path is a dissipative energy, an energy primarily in the form of heat which is not readily put to useful work. The units of the electrical field are newtons/coulomb.