Understanding Gradient Fields and Line Integrals

What is another name for a gradient field?

• Irrotational field

• Divergent field

• Rotational field

• Conservative field

What is a key assumption when using shortcuts in line integrals?

• The field must be rotational

• The field must be simply connected

• The field must be closed

• The field must be divergent

What does path independence imply about a vector field?

• The field is rotational

• The field is divergent

• The field is conservative

• The field is closed

How can you calculate work along a curve in a gradient field?

• By calculating the curl

• By parameterizing the curve

• By finding the divergence

• By using the potential function

What is the potential function for the vector field (y, x)?

• x + y

• xy

• x^2 + y^2

• x - y

What is the work done around a closed curve in a gradient field?

• Equal to the area enclosed

• Zero

• Equal to the perimeter

• Depends on the path

What defines a simple closed curve?

• A curve that crosses itself

• A curve with no inside or outside

• A curve with multiple endpoints

• A curve that does not cross itself

What is the result of a line integral over a full circle in a gradient field?

• Equal to the circumference

• Zero

• Equal to the diameter

• Depends on the radius

How do you find the integral over a curve if the vector field is not a gradient field?

• Use the potential function

• Parameterize the curve

• Use the curl

• Use the divergence theorem

What theorem can be used as a shortcut for closed curves in non-gradient fields?

• Green's theorem

• Gauss' theorem

• Fundamental theorem of calculus

• Stokes' theorem