25  Partial change and the gradient vector

\[ \newcommand{\dnorm}{\text{dnorm}} \newcommand{\pnorm}{\text{pnorm}} \newcommand{\recip}{\text{recip}} \]

Reading questions

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

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 \(\nabla f(x,y)\) 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 \(\nabla f(x,y)\) 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 \(\nabla g(x,y,z)\) evaluated at a point on the contour balloon be related to the surface of the balloon?

  1. 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 \(\nabla g(x, y, z)\) is, as the notation suggests, a function with three inputs, x, y, and z. Write \(\nabla g(x, y, z)\) 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: \[\left(\begin{array}{c}\partial_x g(2, 1.5, 3)\\ \partial_y g(2, 1.5, 3)\\ \partial_z g(2, 1.5, 3)\end{array}\right). \tag{25.1}\]

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.

No answers yet collected