Question

Give a geometrical interpretation of the directional derivative \(\partial f\left(x_{0}, y_{0}\right) / \partial \Phi\) of a function of two variables.

Short answer

Answer: The directional derivative can be seen as the projection of the gradient vector onto the direction vector. It represents the rate of change of the function along the direction of the vector and varies depending on the angle between the gradient and the direction vector. When they are parallel, the directional derivative is at its maximum; when orthogonal, the directional derivative is 0; and when opposite, the directional derivative is at its minimum.

Step-by-step solution

Understanding the concept of directional derivatives

Directional derivatives are the rates of change of a scalar function along a particular direction. It helps us understand how much the function changes when we move in a specific direction in the two-dimensional xy-plane.

Review the formula for directional derivatives

Given a function \(f(x, y)\) and a point \((x_0, y_0)\), the directional derivative of the function at this point along the vector \(\vec{\Phi}\) is: $$\frac{\partial f(x_0, y_0)}{\partial \Phi} = \nabla f(x_0, y_0) \cdot \vec{\Phi_u}$$ where \(\nabla f(x_0, y_0)\) denotes the gradient of the function at this point, and \(\vec{\Phi_u}\) is the unit vector in the direction of \(\vec{\Phi}\).

Discuss the role of the dot product in the formula

The dot product \(\nabla f(x_0, y_0) \cdot \vec{\Phi_u}\) can be written as: $$|\nabla f(x_0, y_0)||\vec{\Phi_u}|\cos{\theta}$$ where \(\theta\) is the angle between the gradient and the unit vector in the direction of \(\vec{\Phi}\), and \(|\nabla f(x_0, y_0)|\) and \(|\vec{\Phi_u}|\) are the magnitudes of the gradient and the unit vector, respectively. Since \(\vec{\Phi_u}\) is a unit vector, its magnitude is 1, and the dot product is equal to the product of the magnitude of the gradient and the cosine of the angle between the gradient and the direction of \(\vec{\Phi}\).

Geometrical interpretation of the directional derivative

The directional derivative can be seen as the projection of the gradient vector onto the direction vector \(\vec{\Phi}\). It represents the rate of change of the function along the direction of the vector \(\vec{\Phi}\). When the angle between the gradient and the direction vector is 0 (meaning they are parallel), the directional derivative is at its maximum and equals the magnitude of the gradient. If the angle is 90 degrees (meaning they are orthogonal), the directional derivative is 0, which indicates there is no change in the function along the direction of the vector \(\vec{\Phi}\). If the angle is 180 degrees (meaning they are opposite), the directional derivative is at its minimum and equals the negative magnitude of the gradient.

Key concepts

Understanding the Gradient Vector

In calculus, the gradient vector is a key concept when dealing with multivariable functions. Simply put, it points in the direction of the greatest rate of increase of a function. If you visualize a mountainous terrain, the gradient is like the direction in which you climb upward most steeply.

Mathematically, for a function \(f(x, y)\), the gradient at any point \((x_0, y_0)\) is denoted by \(abla f(x_0, y_0)\). This is a vector comprised of partial derivatives and can be represented as \((\frac{\partial f}{\partial x}, \frac{\partial f}{\partial y})\).

Some important characteristics of the gradient vector are:

• The direction of the gradient vector indicates the path of steepest ascent.
• The magnitude of the gradient vector indicates how steep the ascent is.
• If the gradient vector is zero, the point is either a local maximum, local minimum, or a saddle point.

Understanding these attributes helps in appreciating how the gradient influences the directional derivative.

The Role of the Dot Product in Directional Derivatives

The dot product is a fundamental concept in vector mathematics, providing a way to calculate how much one vector goes in the direction of another. When applied to the formula for directional derivatives, it links the gradient vector and direction vector \(\vec{\Phi}\).

For a function \(f\), the directional derivative \(\frac{\partial f}{\partial \Phi}\) at a point, explained in terms of the dot product, is \( abla f(x_0, y_0) \cdot \vec{\Phi_u} \), where \(\vec{\Phi_u}\) is the unit direction vector. The dot product can be further broken down into:

• \(|abla f(x_0, y_0)|\): the magnitude of the gradient vector.
• \(|\vec{\Phi_u}|\): the magnitude of the unit vector, which is always 1.
• \(\cos{\theta}\): the cosine of the angle \(\theta\) between the gradient vector and the direction vector.

These components come together to show how much the gradient vector contributes in the direction of \(\vec{\Phi}\). When the gradient and direction vectors are parallel, the dot product reaches its maximum, equating the magnitude of the gradient. Conversely, when they are perpendicular, the dot product becomes zero.

Exploring Rates of Change

Rates of change are central to understanding derivatives, especially directional derivatives. In the context of a function of several variables, the rate of change along a specific direction gives valuable insights.

The directional derivative represents the rate at which a function changes at a certain point, in a specific direction. It reveals how the function \(f(x, y)\) varies when you move in the direction specified by a vector \(\vec{\Phi}\).

Key points to understand about rates of change in directional derivatives include:

• Maximum rate of change occurs in the direction of the gradient vector.
• No change occurs when the point moves orthogonally to the gradient vector.
• Negative change, indicating a decrease, occurs when moving in the opposite direction to the gradient.

The concept of rate of change is essential in fields like physics, engineering, or any domain where predicting and controlling changes is critical. Mastering the directional derivative as a tool for analyzing rates of change sharpens one's analytical skills in these fields.