Chapter 3: 1 P (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.
$f (r) = A \cdot r + 3$ , where A is a give vector.

Short Answer

Expert verified

The function f(r) is not linear.

Step by step solution

Equations for the linear function

A function of a vector, say , 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 isand it is $f (r) = A \cdot r + 3$ to be checked whether the function is linear.

The Left-Hand Side (LHS) of the equation $f (r_{1} + r_{2}) = f (r_{1}) + f (r_{2})$ .

$f (r_{1} + r_{2}) = A \cdot (r_{1} + r_{2}) + 3$

The Right-Hand Side (RHS) of the equation $f (r_{1} + r_{2}) = f (r_{1}) + f (r_{2})$ .

$\begin{array}{rcl} f (r_{1}) + f (r_{2}) & = & A \cdot r_{1} + 3 + A \cdot r_{2} + 3 \\ = & A \cdot (r_{1} + r_{2}) + 6 \end{array}$

As LHS is not equal to RHS, so is does not satisfy the first equation. Therefore, the function is not linear.

Check for the second equation

The Left-Hand Side (LHS) of the equation $f (a r) = a f (r)$ .

$f (a r) = a A \cdot r + 3$

The Right-Hand Side (RHS) of the equation $f (a r) = a f (r)$ .

$\begin{array}{rcl} a f (r) & = & a (A \cdot r + 3) \\ = & a A \cdot r + 3 a \end{array}$

Thus, the function does not satisfy both the equations.

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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.
$f (r) = A \cdot r + 3$ , where A is a give vector.

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.f(r)=A·r+3, where A is a give vector.

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

Scientific Method Physics

Circular Motion and Gravitation

Electric Charge Field and Potential

Thermodynamics

Electromagnetism

Rotational Dynamics

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.
$f (r) = A \cdot r + 3$ , where A is a give vector.