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Chapter 13: Problem 1

Explain how a directional derivative is formed from the two partial derivatives \(f_{x}\) and \(f_{y}\).

Short Answer

Expert verified
Answer: A directional derivative is formed from the two partial derivatives \(f_{x}\) and \(f_{y}\) by taking the dot product of the gradient vector \(\nabla f = \left\langle f_{x}, f_{y} \right\rangle\) and the given unit vector \(\mathbf{u} = \left\langle u_x, u_y \right\rangle\). The formula for the directional derivative is: \(D_{\mathbf{u}}f = f_{x}u_x + f_{y}u_y\). This represents the rate of change of the function in a specified direction, combining the individual rates of change with respect to \(x\) and \(y\).

Step by step solution

01

Define directional derivative

A directional derivative denoted as \(D_{\mathbf{u}}f\) is the rate at which a function \(f\) changes in the direction of a given unit vector \(\mathbf{u}\). It is a derivative of \(f\) that is influenced by the specific direction rather than just a single variable.
02

Define partial derivatives

Partial derivatives, denoted as \(f_{x}\) and \(f_{y}\), are the rates at which a function \(f\) changes with respect to one of its input variables, \(x\) and \(y\) respectively, while keeping the other input variables constant.
03

Introduce the gradient vector

The gradient vector, denoted as \(\nabla f\), is a vector that contains all the partial derivatives of a scalar function \(f\). In two dimensions, the gradient vector is: \(\nabla f = \left\langle f_{x}, f_{y} \right\rangle\)
04

Relate the gradient vector to the directional derivative

The gradient vector is used to find the directional derivative. The directional derivative is the dot product of the gradient vector and the unit vector in the direction of interest. Mathematically it is written as: \(D_{\mathbf{u}}f = \nabla f \cdot \mathbf{u}\)
05

Show the formula for the directional derivative from the partial derivatives

From the previous step, we can calculate the directional derivative from the partial derivatives by taking the dot product of the gradient vector and the unit vector: \(D_{\mathbf{u}}f = \nabla f\cdot\mathbf{u} = \left\langle f_{x}, f_{y} \right\rangle \cdot \left\langle u_x, u_y \right\rangle = f_{x}u_x + f_{y}u_y\) So, the directional derivative is formed by the linear combination of the two partial derivatives, \(f_{x}\) and \(f_{y}\), scaled by the components of the given unit vector \(\mathbf{u}\).

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