/*! This file is auto-generated */ .wp-block-button__link{color:#fff;background-color:#32373c;border-radius:9999px;box-shadow:none;text-decoration:none;padding:calc(.667em + 2px) calc(1.333em + 2px);font-size:1.125em}.wp-block-file__button{background:#32373c;color:#fff;text-decoration:none} Problem 52 Define the gradient of a functio... [FREE SOLUTION] | 91影视

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Chapter 11: Problem 52

Define the gradient of a function of two variables. State the properties of the gradient.

Short Answer

Expert verified
The gradient of a function of two variables \(f(x, y)\) is a vector that points in the direction of the greatest rate of increase of function at a point. It is represented by the vector derivative of the function, \(\nabla f = \frac{\partial f}{\partial x}i + \frac{\partial f}{\partial y}j\). The key properties are: it always points in the direction of the greatest rate of increase of the function; its magnitude gives the rate of increase in the direction of the gradient; it is orthogonal to the level curves of the function at a point.

Step by step solution

01

Define Gradient

Consider a function of two variables, \(f(x, y)\). The gradient of this function at a point \((x, y)\) is a vector that points in the direction of the greatest rate of increase of the function at that point. It is represented by the vector derivative of the function, \(\nabla f\), and is typically computed as \(\nabla f = \frac{\partial f}{\partial x}i + \frac{\partial f}{\partial y}j\), where \(i\) and \(j\) are the unit vectors in the x and y directions, respectively, and \(\frac{\partial f}{\partial x}\) and \(\frac{\partial f}{\partial y}\) are the partial derivatives of the function with respect to x and y.
02

Properties of the Gradient

1. The gradient always points in the direction of the greatest rate of increase of the function. \n2. The magnitude of the gradient vector gives the rate of increase of the function in the direction of the gradient. \n3. The direction of the gradient is always orthogonal to the level curves of the function at a particular point.
03

Example

For example, let's consider the function \(f(x, y) = x^{2} + y^{2}\). The partial derivatives are \(\frac{\partial f}{\partial x} = 2x\) and \(\frac{\partial f}{\partial y} = 2y\). So, the gradient vector at any point \((x, y)\) would be \(2xi + 2yj\). This points in the direction of steepest ascent, and its magnitude \(2\sqrt{x^{2} + y^{2}}\) gives the rate of ascent in that direction.

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Most popular questions from this chapter

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