Understanding Divergence in Vector Fields

The video tutorial explores the concept of divergence in vector fields, explaining how it measures the rate of change inward or outward. It covers positive, negative, and zero divergence, using examples to illustrate each case. Positive divergence indicates outward flow, like expanding gas, while negative divergence shows inward flow, like compressed gas. Zero divergence means no net flow, indicating an incompressible field. The tutorial includes visual aids, such as sketching circles in vector fields, to help understand these concepts.

What does the divergence of a vector field measure?

• The rate of change of temperature

• The direction of a magnetic field

• The rate of change inward or outward

• The speed of a moving object

If the divergence of a vector field is positive, what does it indicate?

• The vector field is compressing

• The vector field is expanding

• The vector field is stationary

• The vector field is rotating

In a vector field with positive divergence, what is true about the flow rates?

• Flow rate inward is greater than outward

• Flow rate outward is greater than inward

• Flow rate is zero

• Flow rate inward equals outward

What does a negative divergence in a vector field signify?

• The vector field is rotating

• The vector field is compressing

• The vector field is stationary

• The vector field is expanding

In a vector field with negative divergence, what is true about the flow rates?

• Flow rate is zero

• Flow rate outward is greater than inward

• Flow rate inward is greater than outward

• Flow rate inward equals outward

What does zero divergence in a vector field indicate?

• The vector field is rotating

• The vector field is expanding

• The vector field is compressing

• The vector field is incompressible

In a vector field with zero divergence, what is true about the flow rates?

• Flow rate is zero

• Flow rate inward equals outward

• Flow rate outward is greater than inward

• Flow rate inward is greater than outward

Which of the following is true for a vector field with zero divergence?

• The vector field is compressing

• The vector field is expanding

• The net flow rate is zero

• The flow rate is always inward

In the complex example of negative divergence, what is consistent across different points?

• The flow rate inward is always greater

• The flow rate outward is always greater

• The flow rate is always zero

• The flow rate is always equal

What happens to the flow rate in a vector field with negative divergence?

• It increases inward

• It increases outward

• It decreases inward

• It remains constant