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Chapter 15: Problem 2

How do you compute the gradient of the functions \(f(x, y)\) and \(f(x, y, z) ?\)

Short Answer

Expert verified
Answer: The gradient of a multivariable function is a vector that points in the direction of the steepest ascent (or greatest increase) of the function at any point in its domain. It is represented by the symbol '鈭�' (nabla) followed by the function. To compute the gradient for a function of two variables, \(f(x, y)\), we calculate the partial derivatives with respect to both variables \(x\) and \(y\), and the gradient is given by the vector \(\nabla f(x, y) = \left(\frac{\partial f}{\partial x}, \frac{\partial f}{\_partial y}\right)\). To compute the gradient for a function of three variables, \(f(x, y, z)\), we calculate the partial derivatives with respect to all three variables \(x\), \(y\), and \(z\), and the gradient is given by the vector \(\nabla f(x, y, z) = \left(\frac{\partial f}{\partial x}, \frac{\partial f}{\partial y}, \frac{\partial f}{\partial z}\right)\).

Step by step solution

01

Understand the gradient vector

The gradient of a function is a vector that points in the direction of the steepest ascent (or greatest increase) of the function at any point in its domain. It is represented by the symbol '鈭�' (called nabla) followed by the function, i.e., \(\nabla f\). The gradient vector is a vector of partial derivatives, where each component corresponds to the partial derivative of the function with respect to a specific variable.
02

Compute the gradient for a function of two variables \(f(x, y)\)

To compute the gradient of a function \(f(x, y)\), we need to calculate the partial derivatives of \(f\) with respect to both variables \(x\) and \(y\). These partial derivatives are denoted by \(\frac{\partial f}{\partial x}\) and \(\frac{\partial f}{\partial y}\). The gradient of \(f(x, y)\) is then given by the vector \(\nabla f(x, y) = \left(\frac{\partial f}{\partial x}, \frac{\partial f}{\partial y}\right)\).
03

Compute the gradient for a function of three variables \(f(x, y, z)\)

To compute the gradient of a function \(f(x, y, z)\), we need to calculate the partial derivatives of \(f\) with respect to all three variables \(x\), \(y\), and \(z\). These partial derivatives are denoted by \(\frac{\partial f}{\partial x}\), \(\frac{\partial f}{\partial y}\), and \(\frac{\partial f}{\partial z}\). The gradient of \(f(x, y, z)\) is then given by the vector \(\nabla f(x, y, z) = \left(\frac{\partial f}{\partial x}, \frac{\partial f}{\partial y}, \frac{\partial f}{\partial z}\right)\).

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Key Concepts

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

Partial Derivatives
When we talk about partial derivatives, we're discussing the rate at which a function changes with respect to one variable while keeping all other variables constant. This concept is pivotal in understanding how functions of several variables behave. Unlike ordinary derivatives, partial derivatives are calculated by differentiating with respect to one variable at a time.
  • Symbol: The symbol for a partial derivative is \( \frac{\partial}{\partial} \), a slanted 'd' representing the change with respect to a chosen variable.
  • Calculation: For a function \( f(x, y) \), the partial derivative with respect to \( x \) is \( \frac{\partial f}{\partial x} \), and with respect to \( y \) is \( \frac{\partial f}{\partial y} \).
Partial derivatives are the building blocks of the gradient vector, revealing how each variable individually influences the function. This allows us to analyze and predict the behavior of complex systems involving multiple variables, such as economic models or physical phenomena.
In essence, partial derivatives help identify the sensitivity of a function to changes in each of its variables.
Functions of Multiple Variables
Functions of multiple variables are crucial in expressing relationships where more than one input factor affects an outcome. For instance, temperature might depend on both latitude and altitude, requiring a function that incorporates both variables.
  • Form: A function of two variables can be noted as \( f(x, y) \), indicating that \( f \) depends on both \( x \) and \( y \).
  • Three Variables: Extending to three variables, we write functions as \( f(x, y, z) \), meaning the output is affected by all three inputs.
Each variable in the function can influence the result in unique ways, making it essential to explore how they interact. These functions appear in many fields, including physics for modeling forces, economics for market analysis, and engineering for system design.
Understanding such functions aids in decoding the intricacies of systems where more than one factor plays a significant role.
Vector Calculus
Vector calculus is an extension of calculus that deals with vector fields instead of scalar fields. It involves studying vectors that may vary with position, and it forms the foundation for understanding phenomena like fluid flow and electromagnetic fields.
  • Gradient: The gradient vector, denoted as \( abla f \), is a key concept here. It points in the direction of the greatest rate of increase of the function and indicates how variables need to be adjusted to increase the function's value.
  • Interpretation: In two dimensions, this vector can be depicted in terms of slopes, while in three dimensions, it represents a more complex relationship between inputs and outputs.
Using vector calculus, we can compute the flow of heat, evaluate economic changes, and even understand gravitational fields. It's applied extensively in various fields of science and engineering, making it an indispensable tool for professionals and researchers working with multi-dimensional data.
The connections established through vector calculus provide powerful insights into the depths of complex systems across various domains.

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Most popular questions from this chapter

Use Lagrange multipliers in the following problems. When the constraint curve is unbounded, explain why you have found an absolute maximum or minimum value. Minimum distance to a cone Find the points on the cone \(z^{2}=x^{2}+y^{2}\) closest to the point (1,2,0)

Graph several level curves of the following functions using the given window. Label at least two level curves with their z-values. $$z=x^{2}+y^{2} ;[-4,4] \times[-4,4]$$

Use what you learned about surfaces in Sections 13.5 and 13.6 to sketch a graph of the following functions. In each case, identify the surface and state the domain and range of the function. $$g(x, y)=\sqrt{16-4 x^{2}}$$

Find the domain of the following functions. If possible, give a description of the domains (for example, all points outside a sphere of radius 1 centered at the origin). $$f(w, x, y, z)=\sqrt{1-w^{2}-x^{2}-y^{2}-z^{2}}$$

Find the absolute maximum and minimum values of the following functions over the given regions \(R .\) \(f(x, y)=2 x^{2}-4 x+3 y^{2}+2\) \(R=\left\\{(x, y):(x-1)^{2}+y^{2} \leq 1\right\\}\) (This is Exercise 51 Section \(15.7 .)\)
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