Chapter 3: 2P (page 82)

Question: Are the following linear functions? Prove your conclusions by showing that f(r)satisfies both of the equations (7.1) or that it does not satisfy at least one of them.
2. $f (r) = A \cdot (r - k z)$ .

Short Answer

Expert verified

The function f(r) is linear.

Step by step solution

Equations for the linear function

A function of a vector, say f(r), is called linear if, $f (r_{1} + r_{2}) = f (r_{1}) + f (r_{2})$ , and $f (a r) = a f (r)$ exists, where a is a scalar.

Check for the first equation

The given function is $f (r) = A \cdot (r - k z)$ and it is to be checked whether the function is linear.

The given function is re-written as $f (r) = A \cdot (r - k (k \cdot r))$ .

Find the first equation of linearity.

$\begin{array}{rcl} f (r_{1} + r_{2}) & = & A \cdot ((r_{1} + r_{2}) - k (k \cdot (r_{1} + r_{2}))) \\ = & A \cdot (r_{1} - k (k \cdot r_{1})) + A \cdot (r_{2} - k (k \cdot r_{2})) \\ = & f (r_{1}) + f (r_{2}) \end{array}$

The function satisfies the first equation.

Check for the second equation

The second equation of linearity.

$\begin{array}{rcl} f (a r) & = & A \cdot (a r - k (k \cdot a r)) \\ = & a A \cdot (r - k (k \cdot r)) \\ = & a f (r) \end{array}$

Thus, the function satisfies the second equation.

Since, the function satisfies both the equation hence, the function is linear.

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Question: Are the following linear functions? Prove your conclusions by showing that f(r)satisfies both of the equations (7.1) or that it does not satisfy at least one of them.
2. $f (r) = A \cdot (r - k z)$ .

Short Answer

Step by step solution

Equations for the linear function

Check for the first equation

Check for the second equation

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Question: Are the following linear functions? Prove your conclusions by showing that f(r)satisfies both of the equations (7.1) or that it does not satisfy at least one of them.2.f(r)=A·(r-kz).

Short Answer

Step by step solution

Equations for the linear function

Check for the first equation

Check for the second equation

One App. One Place for Learning.

Most popular questions from this chapter

Recommended explanations on Physics Textbooks

Medical Physics

Rotational Dynamics

Wave Optics

Fields in Physics

Translational Dynamics

Measurements

Study anywhere. Anytime. Across all devices.

Question: Are the following linear functions? Prove your conclusions by showing that f(r)satisfies both of the equations (7.1) or that it does not satisfy at least one of them.
2. $f (r) = A \cdot (r - k z)$ .