Question

Given a function \(f\), explain the relationship between the gradient and the level curves of \(f\).

Short answer

Based on the given step-by-step solution, the relationship between the gradient and level curves of a function is that the gradient vector is always orthogonal (perpendicular) to the level curve at any point on the curve. This is due to the gradient vector representing the direction of the fastest increase in the function, while the level curve has a constant value, with no change in the function value.

Step-by-step solution

Define Level Curve and Gradient

A level curve (also called contour) of a function \(f(x, y)\) is a curve along which the function has a constant value. In other words, for a given constant \(c\), a level curve is the set of all points \((x, y)\) such that \(f(x, y) = c\). They are used to visualize how the function values change in the \(xy\)-plane. The gradient of a function \(f(x, y)\) is a vector field, represented by the symbol \(\nabla f\), and is composed of the partial derivatives of the function with respect to its variables, \(x\) and \(y\): \[\nabla f = \left( \frac{\partial f}{\partial x}, \frac{\partial f}{\partial y} \right)\] It is the direction and rate in which \(f\) is changing the fastest.

Tangent and Normal Vectors to Level Curve

At any point along the level curve, there is a tangent vector and a normal vector. The tangent vector is parallel to the curve, represents the direction of the curve, and is orthogonal to the level gradient. The normal vector is perpendicular to the tangent vector and points in the direction of the gradient.

Relationship between Gradient and Level Curves

The relationship between the gradient and level curves of a function \(f\) is found by using the concept of tangent and normal vectors. The gradient vector is always orthogonal (perpendicular) to the level curve passing through the point \((x, y)\). This is because the gradient represents the direction of the fastest increase in the function, while the level curve has a constant value (no change in \(f\)). The level curve is essentially a path of "equal slope" or "equal height" in the function's surface. In summary, the gradient vector of a function is always perpendicular to level curves at any point on the curve.

Key concepts

Gradient Vector

The gradient vector is a crucial concept when studying multivariable functions. It represents how a function changes at any given point. For a function \(f(x, y)\), the gradient is written as \(abla f = \left( \frac{\partial f}{\partial x}, \frac{\partial f}{\partial y} \right)\). This vector shows the direction in which the function increases most rapidly.

• Direction of Steepest Ascent: The gradient points towards the steepest ascent of the function. It's like climbing a hill where you follow the path that goes upwards the most quickly.
• Magnitude and Rate: The length of the gradient vector illustrates how fast the function is growing in that direction.

Understanding the gradient vector helps in analyzing how a function behaves across different points in its domain.

Partial Derivatives

Partial derivatives are the building blocks of the gradient vector. They measure how a function changes as one variable changes, keeping the other variable constant. For a function \(f(x, y)\), the partial derivatives are:

• \(\frac{\partial f}{\partial x}\): The rate of change of \(f\) with respect to \(x\), holding \(y\) constant.
• \(\frac{\partial f}{\partial y}\): The rate of change of \(f\) with respect to \(y\), holding \(x\) constant.

Partial derivatives help us understand how the function's value shifts when we tweak just one variable. This is vital in constructing the gradient vector, which guides us in determining where the function increases swiftly. Calculating these derivatives provides insights into how functions respond to small changes in their input variables.

Tangent and Normal Vectors

When examining level curves, tangent and normal vectors play a pivotal role. A level curve of a function \(f(x, y)\) is a path where the function maintains a constant value. At any point on this curve, tangent and normal vectors illustrate essential geometrical properties.

• Tangent Vector: This vector is aligned with the curve, representing the curve's direction at that point. It is always perpendicular to the gradient vector.
• Normal Vector: This vector is perpendicular to the tangent vector and is aligned with the gradient vector, showcasing the direction of steepest ascent.
• Gradient and Curves Relationship: The gradient vector being orthogonal to the level curve means it's perpendicular, marking the point of no change in function value along the curve.

Understanding these vectors helps in visualizing how functions behave in 3D space, providing valuable insights into the nature of multivariable functions.