Chapter Overview¶
Where We Are¶
Chapter 8 introduced the coordinates used to represent, and the distributions used to sample, random vectors. In particular, Sections 8.2 and 8.3 introduced scalar-valued functions of multiple variables.
Where We’re Going¶
In this chapter, we will study how to differentiate functions of multiple variables. We will use these tools to find maxima and minima of surfaces, and to find modes of multivariate distributions.
We will:
Define a partial derivative as a derivative of a cross-section of a surface and will show how to use partial derivatives to build linear approximations to surfaces using their tangent planes (see Section 9.1).
Show how to summarize the linearization with gradients (see Section 9.2). We will:
Practice computing gradients
Show how to recover directional derivatives from gradients
Show that the gradient vector points in the direction of steepest ascent, and is perpendicular to the level sets.
Combine these ideas to optimize surfaces. We will use gradients to:
Optimize surfaces without any constraints on the inputs by setting the gradient to zero (see Section 9.3).
Optimize surfaces iteratively by stepping in the direction of the gradient, that is, by gradient ascent (see Section 9.3).