91Ó°ÊÓ

The gradient of a function is like a compass pointing in the direction of the steepest ascent of a hill. For a two-variable function such as
\(f(x, y)\)
, the gradient is a vector that combines the rates of change with respect to each variable. This rate of change for each variable is found by calculating the partial derivatives.

Imagine slicing the hill horizontally at the point of interest and analyzing the slope – that's what each partial derivative represents. Combine these separate slopes into a vector, and you've got the gradient. In our exercise, we found that the gradient of
\(f(x, y)=3-\frac{x}{3}-\frac{y}{2}\)
is
\(abla f(x, y) = \left[-\frac{1}{3}, -\frac{1}{2}\right]\)
which provides information about how
\(f\)
increases or decreases around the point (3,2).

Partial derivatives are the key ingredients in a gradient recipe. They are like the x-ray vision for each variable, where you focus on one while treating all others as constants. When you partially differentiate
\(f(x, y)\)
with respect to
\(x\)
, you simply track how
\(f\)
changes as
\(x\)
wiggles while
\(y\)
stands still. Flip the roles for
\(y\)
's partial derivative.

To do this, consider the function
\(f(x, y)=3-\frac{x}{3}-\frac{y}{2}\)
, we calculate two partial derivatives: one with respect to
\(x\)
and another with respect to
\(y\)
. In our case, these are
\(-\frac{1}{3}\)
and
\(-\frac{1}{2}\)
, respectively. The gradient is merely these partials neatly wrapped up in a vector.

A normalized direction vector is the humble GPS for the gradient, telling it where to go but keeping it grounded so it doesn't get too large. To normalize a vector, you divide it by its magnitude, scaling it down to a unit vector. A unit vector has a length of one, which is perfect for indicating direction without implying intensity.

In the context of our exercise, two vectors
\(\mathbf{v}\)
are given, to be normalized. The process entails dividing each vector by its magnitude to get the unit vectors
\(\mathbf{u}\)
. For vector
\(\mathbf{i} + \mathbf{j}\)
, its magnitude is
\(\sqrt{2}\)
, and hence the normalized direction vector is
\(\frac{\mathbf{i} + \mathbf{j}}{\sqrt{2}}\)
. This vector points in the same direction as
\(\mathbf{v}\)
but with a reduced, uniform length.

The dot product is like a handshake between two vectors, measuring how much they agree on their direction and magnitude. It is the algebraic form of multiplying vectors which results in a scalar, showing the projection of one vector onto another. For two vectors
\(\mathbf{a}\)
and
\(\mathbf{b}\)
, the dot product is
\(\mathbf{a} \cdot \mathbf{b} = a_xb_x + a_yb_y\)
, adding together the products of their corresponding components.

In our directional derivative calculation, we used the dot product to measure how much the gradient of
\(f\)
aligns with our unit direction vectors
\(\mathbf{u}\)
. This quantifies the rate of change of
\(f\)
in the direction of
\(\mathbf{u}\)
at the point (3,2). It's a valuable tool for understanding how functions change not just overall, but in specific directions.