25  Partial change and the gradient vector

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Reading questions

Reading question 25.1 Which of the following is the correct interpretation of $\mathrm{\nabla }f()$?

The function that is the finite-difference derivative of $f()$.

The function that is the anti-difference of $f()$.

The function that is the gradient of $f()$.

The function that is the inverse of $f()$.

question id: nabla-meaning

Reading question 25.2 On a contour plot of a function $f(x,y)$, an argmax will be located inside the set of concentric contours that show the sides of a “mountain.” At this argmax, what will the value of the gradient vector $\mathrm{\nabla }f(x,y)$ be. (Hint: You don’t need to know a specific $f()$, just that you are evaluating the gradient at an argmax.)

Reading question 25.3 For a function $f(x,y)$ with two inputs, a contour plot shows where in $(x,y)$ space the function output takes on specific values. Imagine a single contour, perhaps one with a circular shape. How will the gradient vector $\mathrm{\nabla }f(x,y)$ evaluated at a point on the contour be related to the curve of the contour going through that point.

Reading question 25.4 For a function $g(x,y,z)$ the analog of a contour is three-dimensional. For instance, the 3-d “contour” can be likened to the surface of a balloon positioned in an $(x,y,z)$ space.

1. How will the gradient vector $\mathrm{\nabla }g(x,y,z)$ evaluated at a point on the contour balloon be related to the surface of the balloon?

2. Close enough to a point on the balloon’s surface, it resembles a planar surface (locally). Take a book and lean it up against something else so that the cover is neither vertical nor horizontal. Now put a pencil’s eraser end at a point on the book cover and orient the pencil to be perpendicular to the cover. Is there a unique orientation that achieves this, or are there multiple, different orientations?

Reading question 25.5 The gradient $\mathrm{\nabla }g(x,y,z)$ is, as the notation suggests, a function with three inputs, x, y, and z. Write $\mathrm{\nabla }g(x,y,z)$ as a coordinate triple, where each component of the triple is the partial derivative of $g()$ with respect to one of the inputs.

Evaluating the coordinate triple for a specific input, say $x=2,y=1.5,z=3$, will give three numbers as output. You are acquainted with such triples as representing position in space, e.g. $(2,1.5,3)$. For vectors, however, it is conventional to write the components one on top of the other in a column, like this: $$\begin{array}{}\text{(25.1)}& \left(\begin{array}{c}{\partial }_{x}g(2,1.5,3)\\ {\partial }_{y}g(2,1.5,3)\\ {\partial }_{z}g(2,1.5,3)\end{array}\right).\end{array}$$

It might look at first glance like the three components in Math expression 25.1 are identical to one another. Explain, in typographical terms, what’s different about the components.

Note: Our physical representation of a position in space is a “point,” drawn as a dot. For vectors, our physical representation is a pencil or arrow with the sharp end pointing forward.

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