Question

Starting from the point \((1,1),\) in what direction does the function \(\phi=x^{2}-y^{2}+2 x y\) decrease most rapidly?

Short answer

The function decreases most rapidly in the direction \((-1, 0)\).

Step-by-step solution

- Find the gradient of the function

The gradient of the function \(\phi(x, y) = x^2 - y^2 + 2xy\) is given by \(abla\phi = \left(\frac{\partial \phi}{\partial x}, \frac{\partial \phi}{\partial y}\right)\). First, compute the partial derivatives: \(\frac{\partial \phi}{\partial x} = 2x + 2y\) and \(\frac{\partial \phi}{\partial y} = -2y + 2x\).

- Evaluate the gradient at the given point

Now, evaluate the gradient at the point \((1, 1)\). This yields: \(\left.\frac{\partial \phi}{\partial x}\right|_{(1,1)} = 2(1) + 2(1) = 4\), and \(\left.\frac{\partial \phi}{\partial y}\right|_{(1,1)} = -2(1) + 2(1) = 0\). Hence, the gradient at the point \((1, 1)\) is \(abla\phi(1, 1) = (4, 0)\).

- Determine the direction of most rapid decrease

The direction of the most rapid decrease of the function is given by the negative of the gradient vector. Therefore, the direction is opposite to \(abla\phi(1, 1)\). This results in \(-abla\phi(1, 1) = (-4, 0)\).

- Normalize the direction vector

Normalize the direction vector \((-4, 0)\) to get the unit direction vector. The magnitude of \((-4, 0)\) is \(\sqrt{(-4)^2 + 0^2} = 4\). Thus, the unit direction vector is \(\frac{(-4, 0)}{4} = (-1, 0)\).

Key concepts

gradient

The gradient is a crucial concept in multivariable calculus and optimization. It is a vector that points in the direction of the steepest ascent of a function. For a function \( \phi(x,y) \) with two variables, the gradient \( \abla \phi \) consists of the partial derivatives of the function with respect to each variable. In mathematical notation, this is written as: \[ \abla \phi = \left(\frac{\partial \phi}{\partial x}, \frac{\partial \phi}{\partial y}\right) \]

The gradient vector tells us how the function changes as we move in different directions. For example, in the exercise, the gradient of \( \phi(x, y) = x^2 - y^2 + 2xy \) at the point \( (1, 1) \) is calculated as:

• \( \frac{\partial \phi}{\partial x} = 2x + 2y \)
• \( \frac{\partial \phi}{\partial y} = -2y + 2x \)

Evaluated at \( (1, 1) \), the gradient becomes \( (4, 0) \). This gradient vector points in the direction of fastest increase of the function.

partial derivatives

Partial derivatives show how a function changes as one variable changes, keeping others constant. They are the building blocks for the gradient. For a function of two variables \( \phi(x, y) \), the partial derivatives with respect to x and y are represented as: \[ \frac{\partial \phi}{\partial x}, \frac{\partial \phi}{\partial y} \]

In the exercise, these derivatives are:

• \( \frac{\partial \phi}{\partial x} = 2x + 2y \)
• \( \frac{\partial \phi}{\partial y} = -2y + 2x \)

When evaluating these at the point \( (1, 1) \), we get:

• \( 2(1) + 2(1) = 4\right|_{(1,1)} \)
• \( -2(1) + 2(1) = 0 \)

This gives us the gradient vector \( (4, 0) \), compiling the partial derivatives into one vector.

direction of descent

The direction of descent is the direction in which a function decreases the fastest. This direction is opposite to the gradient vector. If the gradient points toward the steepest ascent, then the negative gradient points to the steepest descent. In the context of the exercise, the gradient at \( (1, 1) \) is \( (4, 0) \). To find the direction of most rapid decrease, we take the negative of this gradient: \( (-4, 0) \). This means moving in the direction \( (-4, 0) \) will decrease the function value the fastest, starting from the point \( (1, 1) \).

normalization

Normalization is the process of scaling a vector to have a unit length. This simplifies calculations and comparisons. For any vector \( \mathbf{v} = (v_x, v_y) \), the magnitude or length is given by \[ \|\mathbf{v}\| = \sqrt{v_x^2 + v_y^2} \]

To normalize the vector, we divide each component by its magnitude. In the exercise, the direction vector \( (-4, 0) \) has a magnitude of:
\[ \sqrt{(-4)^2 + 0^2} = 4 \]
The normalized form is then:
\[ \frac{(-4, 0)}{4} = (-1, 0) \]
This unit direction vector tells us the precise direction of descent but with a step size of 1.