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Chapter 13: Problem 4

Calculate the gradient field and the Laplacian of each of the following: a. \(f(x, y)=x^{2}+x y\) b. \(f(x, y)=e^{x+y}\) c. \(f(x, y, z)=\frac{x+y}{z}\)

Short Answer

Expert verified
a) Gradient: \( (2x+y, x) \), Laplacian: 2; b) Gradient: \( (e^{x+y}, e^{x+y}) \), Laplacian: 0; c) Gradient: \( \left( \frac{1}{z}, \frac{1}{z}, -\frac{x+y}{z^2} \right) \), Laplacian: \( \frac{2(x+y)}{z^3} \).

Step by step solution

01

Understand the Gradient

The gradient of a function \( f(x, y) \) or \( f(x, y, z) \) is a vector of its first partial derivatives. For two variables, it is \( abla f = \left( \frac{\partial f}{\partial x}, \frac{\partial f}{\partial y} \right) \). For three variables, it is \( abla f = \left( \frac{\partial f}{\partial x}, \frac{\partial f}{\partial y}, \frac{\partial f}{\partial z} \right) \).
02

Calculate the Gradient for each Function

For \( f(x, y) = x^2 + xy \), the gradient is \( abla f = (2x + y, x) \).For \( f(x, y) = e^{x+y} \), the gradient is \( abla f = (e^{x+y}, e^{x+y}) \).For \( f(x, y, z) = \frac{x+y}{z} \), the gradient is \( abla f = \left( \frac{1}{z}, \frac{1}{z}, -\frac{x+y}{z^2} \right) \).
03

Understand the Laplacian

The Laplacian of a function \( f(x, y) \) in two dimensions is \( abla^2 f = \frac{\partial^2 f}{\partial x^2} + \frac{\partial^2 f}{\partial y^2} \). For three variables \( f(x, y, z) \), the Laplacian is \( abla^2 f = \frac{\partial^2 f}{\partial x^2} + \frac{\partial^2 f}{\partial y^2} + \frac{\partial^2 f}{\partial z^2} \).
04

Calculate the Laplacian for each Function

For \( f(x, y) = x^2 + xy \), take the second derivatives: \( \frac{\partial^2 f}{\partial x^2} = 2 \), \( \frac{\partial^2 f}{\partial y^2} = 0 \). Thus, \( abla^2 f = 2 \).For \( f(x, y) = e^{x+y} \), \( \frac{\partial^2 f}{\partial x^2} = 0 \) and \( \frac{\partial^2 f}{\partial y^2} = 0 \), giving \( abla^2 f = 0 \).For \( f(x, y, z) = \frac{x+y}{z} \): \( \frac{\partial^2 f}{\partial x^2} = 0 \), \( \frac{\partial^2 f}{\partial y^2} = 0 \), \( \frac{\partial^2 f}{\partial z^2} = \frac{2(x+y)}{z^3} \), thus \( abla^2 f = \frac{2(x+y)}{z^3} \).

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Key Concepts

These are the key concepts you need to understand to accurately answer the question.

Gradient
The gradient is a fundamental concept in multivariable calculus, acting as a multi-dimensional generalization of the derivative. When we talk about a gradient, we're referring to the vector that represents the direction and rate of the steepest ascent of a function. In simpler terms, it points in the direction where the function rises most rapidly.

Key aspects of the gradient include:
  • Being a vector composed of all first-order partial derivatives of a function.
  • Showing how the function changes in each direction of its variables.
  • For a two-variable function like \( f(x, y) \), the gradient \( abla f = \left( \frac{\partial f}{\partial x}, \frac{\partial f}{\partial y} \right) \).
  • For three variables, as in \( f(x, y, z) \), the gradient extends to \( abla f = \left( \frac{\partial f}{\partial x}, \frac{\partial f}{\partial y}, \frac{\partial f}{\partial z} \right) \).
The gradient provides crucial insights in fields such as optimization, where understanding the direction of fastest increase of a function is vital to finding maximum or minimum values.
Laplacian
The Laplacian is a measure of the rate at which a quantity spreads out from a point, common in fields like physics and engineering, especially for phenomena like heat transfer or fluid dynamics. It's essentially a scalar function that gives us an insight into the concavity or convexity of a function.

Some important points about the Laplacian include:
  • It is obtained by summing the second partial derivatives of a function.
  • In two dimensions, for a function \( f(x, y) \), Laplacian is written as \( abla^2 f = \frac{\partial^2 f}{\partial x^2} + \frac{\partial^2 f}{\partial y^2} \).
  • For three dimensions, the formula extends to \( abla^2 f = \frac{\partial^2 f}{\partial x^2} + \frac{\partial^2 f}{\partial y^2} + \frac{\partial^2 f}{\partial z^2} \).
  • Physically, a positive Laplacian indicates a local minimum (concave upwards), while a negative Laplacian suggests a local maximum (concave downwards).
The Laplacian is pivotal in differential equations and mathematical physics, offering insights into the behavior of dynamic systems.
Partial Derivatives
Partial derivatives are the building blocks for gradients and the Laplacian in multivariable calculus. They measure the change of a function with respect to one of its variables, while keeping all others constant. Understanding how each variable individually influences the function is crucial in modeling and solving real-world problems.

Key insights about partial derivatives are:
  • They are denoted as \( \frac{\partial f}{\partial x} \) for the derivative of \( f \) with respect to \( x \), while treating other variables like \( y \) and \( z \) as constants.
  • Partial derivatives allow us to analyze the sensitivity of a function to small changes in one direction.
  • They are used to compute the gradient and Laplacian, which are higher-order derivatives that account for changes across multiple variables.
  • In practical terms, partial derivatives help in understanding how economic factors, physical forces, or any other elements with multiple influences behave under minor variations.
Mastering partial derivatives prepares you for deeper exploration in calculus, such as finding local extrema or solving multivariable differential equations.

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Most popular questions from this chapter

Plot the scalar field \(f(x, y)=\sin (x y)\) for \(-4 \leq x, y \leq 4\) using Python.

So far, the gradient is defined only for a scalar field, but it is natural to define a gradient for a vector field too. Propose a meaningful definition of \(\nabla v\) and discuss its physical meaning.

Develop a PDE-based model that describes the dynamics of a predator-prey ecosystem with the following assumptions: \- The prey population grows following the logistic growth model. \- The predator population tends to decay naturally without prey. \- Predation (i.e., encounter of prey and predator populations) increases the predator population while it decreases the prey population. \- The predator population tends to move toward areas with high prey density. \- The prey population tends to move away from areas with high predator density. \- Both populations diffuse over space.

Plot the gradient field of \(\mathrm{f}(x, y)=\sin (x y)\) for \(-4 \leq x, y \leq 4\) using Python.
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