Gradient of a Function
Understanding the gradient of a function is pivotal in multivariable calculus. The gradient measures the slope or rate at which the function changes in space, and it is a vector pointing in the direction of the steepest increase. To find the gradient, we compute the partial derivatives with respect to each variable and combine these derivatives into a vector. For instance, consider a function with three variables, f(x, y, z). The gradient of f is a vector given by \( abla f = \langle \frac{\partial f}{\partial x}, \frac{\partial f}{\partial y}, \frac{\partial f}{\partial z} \rangle \), where each component of this vector is a partial derivative of the function with respect to one of its variables.
When evaluating the gradient at a specific point, you substitute the coordinates of that point into each of the partial derivatives. This gives you a vector that describes how steeply the function increases at that particular location and in which direction. For the given function in the exercise, \( f(x, y, z)=\ln(1+x^{2}+y^{2}+z^{2}) \), the gradient at point P(1,1,-1) was calculated as \( abla f(1, 1, -1) = \langle \frac{1}{2}, \frac{1}{2}, -\frac{1}{2} \rangle \).
Partial Derivatives
Partial derivatives are the foundation of the gradient concept. They measure how a function changes as each variable changes, while all other variables are held constant. This is akin to examining the slope of a function in one direction at a time. For a function with several variables, you will have as many partial derivatives as there are variables.
To compute a partial derivative, you differentiate the function with respect to one variable, treating all other variables as constants. In the provided exercise, the partial derivatives of f with respect to x, y, and z are calculated using the chain rule which results in \( \frac{2x}{1+x^2+y^2+z^2} \), \( \frac{2y}{1+x^2+y^2+z^2} \), and \( \frac{2z}{1+x^2+y^2+z^2} \) respectively. These derivatives reflect the rate at which f changes in response to changes in x, y, and z when evaluated at a specific point.
Rate of Change
The rate of change in a specific direction is a critical aspect of understanding how functions behave in space. It can be computed as the dot product of the gradient with a unit vector that represents the direction of interest. The result tells us how fast the function is increasing or decreasing as we move in that direction. If you're looking at the direction of the gradient itself, the rate of change gives you the maximum rate at which the function increases.
In our example, after finding the unit vector for the direction of maximum increase, you can calculate the rate of change in that direction. The rate at P was found to be \( \frac{1}{\sqrt{3}} \). Remember, if you were to choose a direction where the function decreases, the rate of change would be negative, indicating a decline in the function's value as you move in that direction.
Unit Vector
A unit vector is a vector with a length (or magnitude) of 1. It is often used to indicate direction without specifying magnitude. To find a unit vector in the direction of another vector, you divide the original vector by its magnitude, which normalizes it. This process is employed when determining the direction of maximum increase of a function, as the unit vector points in that direction.
In our exercise, the unit vector in the direction of the gradient at P is \( \left\langle \frac{1}{\sqrt{3}}, \frac{1}{\sqrt{3}}, -\frac{1}{\sqrt{3}} \right\rangle \). Importantly, any vector, like the one given in the exercise \( \left\langle \frac{2}{3}, \frac{2}{3}, -\frac{1}{3} \right\rangle \), can also be converted into a unit vector by dividing it by its own magnitude, and it can be used to find the directional derivative, which describes how the function changes in that specific direction.
