What makes a field conservative?

A force is called conservative if the work it does on an object moving from any point A to another point B is always the same, no matter what path is taken. In other words, if this integral is always path-independent.

What do you understand by divergence and curl of a vector give their physical significance?

The divergence of a vector field is a scalar function. The curl of a vector field is a vector field. The curl of a vector field at point P measures the tendency of particles at P to rotate about the axis that points in the direction of the curl at P.

Why does a conservative field have zero curl?

Because by definition the line integral of a conservative vector field is path independent so there is a function f whose exterior derivative is the gradient df. Than the curl is *d(df)=0 because the boundary of the boundary is zero, dd=0.

What is a conservative vector field example?

The vector field F is said to be conservative if it is the gradient of a function. Such a function f is called a potential function for F. Example 1.2. F(x, y, z) = (y2z3,2xyz3,3xy2z2) is conservative, since it is F = ∇f for the function f(x, y, z) = xy2z3.

How do you know if a force is conservative or not?

If the derivative of the y-component of the force with respect to x is equal to the derivative of the x-component of the force with respect to y, the force is a conservative force, which means the path taken for potential energy or work calculations always yields the same results.

Which of the following field is not conservative?

Frictional force is a non-conservative force.

What is divergence of curl of a vector?

The curl of the vector field is defined as: The divergence of a vector field is defined as: The divergence of a vector field represents the extent to which a vector tends to diverge at a point. If the vector converges into a point, then the divergence would be negative.

How do you know if a vector field is conservative?

As mentioned in the context of the gradient theorem, a vector field F is conservative if and only if it has a potential function f with F=∇f. Therefore, if you are given a potential function f or if you can find one, and that potential function is defined everywhere, then there is nothing more to do.

What is the curl of conservative field?

The curl of every conservative field is equal to zero.

What do you understand by conservative field write two examples of conservative field?

A conservative force is a force with the property that the total work done in moving a particle between two points is independent of the path taken. Other examples of conservative forces are: force in elastic spring, electrostatic force between two electric charges, and magnetic force between two magnetic poles.

Which of the following forces is not conservative?

The correct answer is Frictional force. The frictional force is a non-conservative force.

How to determine if a field is conservative or not?

The field is conservative if it is the gradient of a scalar field. That means: F (x,y) = (∂u/∂x,∂u/∂y) The field you gave is indeed x and y components. you could also write it in terms of basis vectors: F = 2yi+2xj. To determine if it is conservative you can find the actual function u such that F = ∇u. But a sufficiency test is…

What is the definition of a conservative vector field?

Recall that the reason a conservative vector field F is called “conservative” is because such vector fields model forces in which energy is conserved. We have shown gravity to be an example of such a force. If we think of vector field F in integral as a gravitational field, then the equation follows.

How do you express the vector v in arbitrary form?

Arbitrary vector v is expressed by the linear combination of the independent vectors a, b, c as follows: where the coefficients va, vb, vc are given by operating the scalar products of a×b, b×c, and c×a to Eq. (1.62) as follows: The vector v is rewritten by substituting Eq. (1.63) into Eq. (1.62) as follows:

Is the vector field of a line independent of the path?

Call the initial point and the terminal point Since F is conservative, there is a potential function for F. By the Fundamental Theorem for Line Integrals, Therefore, and F is independent of path. The vector field is conservative, and therefore independent of path.