Vector Calculus

Step 1: Understanding the Concept:
The question asks to identify the correct graphical representation of the vector field . A vector field assigns a vector (with magnitude and direction) to every point in space. We need to analyze the properties of the given vector field and match them with the figures.
Step 2: Key Formula or Approach:
We analyze the vector field based on its direction and magnitude at different points in the -plane.

• Direction: The direction of the vector is determined by the sign of its components.
• Magnitude: The magnitude of the vector is .

Step 3: Detailed Explanation:
Let's analyze the vector field in detail:
1. Direction of Vectors:

• The vector field has only an (x-component). This means all vectors must be horizontal, pointing either to the right ( ) or to the left ( ). All four figures show horizontal vectors.
• The x-component is .
• When (i.e., above the x-axis), the component is positive, so the vectors should point to the right ( ).
• When (i.e., below the x-axis), the component is negative, so the vectors should point to the left ( ).
• When (i.e., on the x-axis), the component is zero, so the vector is a zero vector (a point).

2. Magnitude of Vectors:

• The magnitude is . This means the length of the vector arrows should be proportional to the distance from the x-axis.
• As we move away from the x-axis in either the positive or negative y-direction, the magnitude increases, so the arrows should become longer.

Now let's examine the options:

• Figure (A):
  • For , arrows point right. (Correct)
  • For , arrows point left. (Correct)
  • As increases, the arrows get longer. (Correct)
  • This figure correctly represents the vector field .
• Figure (B):
  • For , arrows point right. (Incorrect, they should point left).
• Figure (C):
  • For , arrows point left. (Incorrect, they should point right).
• Figure (D):
  • The directions are correct (right for , left for ).
  • However, as increases, the arrows get shorter. This is incorrect, as the magnitude should increase.

Step 4: Final Answer:
Based on the analysis of direction and magnitude, only Figure (A) provides a correct representation of the vector field .