Question

Suppose a long sloping hillside is described by the plane \(z=a x+b y+c,\) where \(a, b,\) and \(c\) are constants. Find the path in the \(x y\) -plane, beginning at \(\left(x_{0}, y_{0}\right),\) that corresponds to the path of steepest ascent on the hillside.

Short answer

Answer: The parametric equation for the path of steepest ascent in the xy-plane, starting from the point (x_0, y_0), for a hillside with height represented by the function z = ax + by + c is given by: \(P(t) = \left( x_0 + t \cdot \frac{a}{\sqrt{a^2 + b^2}}, y_0 + t \cdot \frac{b}{\sqrt{a^2 + b^2}} \right)\).

Step-by-step solution

Compute the gradient vector of the plane

To compute the gradient vector of the plane \(z = ax + by + c\), we need to find the partial derivatives with respect to \(x\) and \(y\). The gradient vector will have the form \(\nabla{z} = (\frac{\partial z}{\partial x}, \frac{\partial z}{\partial y})\). Let's find the partial derivatives: \(\frac{\partial z}{\partial x} = a\) \(\frac{\partial z}{\partial y} = b\) So, the gradient vector is \(\nabla{z} = (a, b)\).

Find the tangent vector of the steepest path (normalize the gradient vector)

The gradient vector points in the direction of steepest ascent, and its magnitude represents the rate of ascent. To find the tangent vector of the steepest path, we will normalize the gradient vector by dividing it by its magnitude. The tangent vector will have unit length and point in the direction of steepest ascent. Let's compute the magnitude and normalize the gradient vector: \(\lVert \nabla{z} \rVert = \sqrt{a^2 + b^2}\) Tangent vector of the steepest path: \(T = \frac{\nabla{z}}{\lVert \nabla{z} \rVert} = \left( \frac{a}{\sqrt{a^2 + b^2}}, \frac{b}{\sqrt{a^2 + b^2}} \right)\)

Write the parametric equation for the path of steepest ascent

Now, we have the tangent vector of the steepest path and we need to find the parametric equation for this path. Let this path be represented by the function \(P(t) = (x(t), y(t))\). We will use the initial point \((x_0, y_0)\) and the tangent vector \(T\) to write the parametric equation for the path: \(x(t) = x_0 + t \cdot \frac{a}{\sqrt{a^2 + b^2}}\) \(y(t) = y_0 + t \cdot \frac{b}{\sqrt{a^2 + b^2}}\) Thus, the path of steepest ascent in the \(xy\)-plane, starting from the point \((x_0, y_0)\), can be represented as: \(P(t) = \left( x_0 + t \cdot \frac{a}{\sqrt{a^2 + b^2}}, y_0 + t \cdot \frac{b}{\sqrt{a^2 + b^2}} \right)\)

Key concepts

Partial Derivatives

Partial derivatives are essential in multivariable calculus, allowing us to study how a function changes as one of its variables changes while keeping the others constant. These derivatives are especially useful in fields like optimization, where we want to understand the behavior of surfaces or planes.

For example, consider a plane described by the equation \(z = ax + by + c\). To find the partial derivatives, we first take the derivative with respect to \(x\) while treating \(y\) as a constant. This gives us \(\frac{\partial z}{\partial x} = a\). Similarly, the derivative with respect to \(y\), treating \(x\) as a constant, results in \(\frac{\partial z}{\partial y} = b\).

Partial derivatives provide the components for the gradient vector, coordinating how small changes in the \(x\) and \(y\)-directions affect the output \(z\). This concept is key in understanding gradient descent and ascent methods, as it lays the groundwork for examining rates of change across multiple dimensions.

Parametric Equation

A parametric equation gives us a way to express mathematical curves or surfaces by using parameters to describe different positions. Instead of just using \(x\), \(y\), and \(z\) as independent variables, we introduce a parameter, often \(t\), which smoothly varies to trace the path or shape.

In the context of optimization and pathfinding, parametric equations simplify the problem of tracing the path of steepest ascent. Using our gradient-derived tangent vector \(T = \left( \frac{a}{\sqrt{a^2 + b^2}}, \frac{b}{\sqrt{a^2 + b^2}} \right)\), we start at an initial point \((x_0, y_0)\) and build our path function as \(P(t) = (x(t), y(t))\).

The equations \(x(t) = x_0 + t \cdot \frac{a}{\sqrt{a^2 + b^2}}\) and \(y(t) = y_0 + t \cdot \frac{b}{\sqrt{a^2 + b^2}}\) parameterize the trajectory of ascent. They provide a consistent way to map out each point along the path as \(t\) increases, keeping us oriented in the right direction.

Steepest Ascent

Steepest ascent refers to finding the direction which yields the maximum rate of increase for a function. This concept is crucial when seeking optimal paths, particularly when navigating surface gradients.

To identify the path of steepest ascent, we first compute the gradient vector \(abla z = (a, b)\), which inherently directs us towards maximum ascent on a plane described by \(z = ax + by + c\). At any point, the gradient vector's direction points to where \(z\) increases most steeply.

The normalization step that follows involves scaling this gradient vector to ensure it has a magnitude (or length) of one, leading to a unit vector. This ensures that while our directions remain consistent, each step along the path is of manageable size. Thus, the normalized vector \(T = \left( \frac{a}{\sqrt{a^2 + b^2}}, \frac{b}{\sqrt{a^2 + b^2}} \right)\) precisely indicates the unit direction for maximum ascent.

• A normalized vector gives a clear, consistent direction irrespective of the original magnitude.
• The larger the components in the gradient, the steeper the ascent in that direction.

By using steepest ascent methods, we navigate through the landscape of a plane or surface efficiently, reaching higher points in an optimal manner.