Dr. Robert Lopez

The Directional Derivative

The directional derivative of a vector field appears in modern treatments of the curvature of a surface, and in several vector identities met in vector calculus. What is this operator, and how can it be implemented in Maple?

The directional derivative of a scalar field is a staple of the multivariate calculus course. In Cartesian coordinates, it is derived from first principles by differentiating the scalar along a line emanating from a fixed point and pointing in a given direction, then evaluating the derivative in the limit at the fixed point. When this limit exists, the directional derivative can be expressed as the dot product between the gradient vector and a unit vector along the given direction.

But does this result hold for vectors, and in nonCartesian coordinates? This webinar explores whether the definition of the directional derivative extends to vectors, and to scalars and vectors in nonCartesian coordinates. Derivations and implementations in Maple are given for polar coordinates, and the connection with the (directional) covariant derivative of a vector is made clear.
Language: English
Duration: 70 Minutes
Related Terms: Directional-derivatives, Curvature, Vector-calculus, Scalar, Cartesian