How do you find the vector field if curl is given?

Let F(x,y,z)=(y,z,x2) on R3. We know that y=∂G3∂y−∂G2∂z,z=∂G1∂z−∂G3∂x,x2=∂G2∂x−∂G1∂y.

What is the curl of a vector field at a point?

In vector calculus, the curl is a vector operator that describes the infinitesimal circulation of a vector field in three-dimensional Euclidean space. The curl at a point in the field is represented by a vector whose length and direction denote the magnitude and axis of the maximum circulation.

How do you interpret the curl of a vector field?

The curl of a vector field measures the tendency for the vector field to swirl around. Imagine that the vector field represents the velocity vectors of water in a lake. If the vector field swirls around, then when we stick a paddle wheel into the water, it will tend to spin.

Is there a vector field such that curl of G?

No. If there was such a vector field, then div(curl G) = 0.

What is the correct representation of curl of a vector Mcq?

The curl of a vector is a vector only. The curl of the resultant vector is also a vector only. Explanation: Curl F = -5z2 j + 2y k. At (1,1,-0.2), Curl F = -0.2 j + 2 k.

What is the difference between Stokes theorem and Green’s theorem?

Stokes’ theorem is a generalization of Green’s theorem from circulation in a planar region to circulation along a surface. Green’s theorem applies only to two-dimensional vector fields and to regions in the two-dimensional plane. Stokes’ theorem generalizes Green’s theorem to three dimensions.

What does Stokes theorem mean?

The Stoke’s theorem states that “the surface integral of the curl of a function over a surface bounded by a closed surface is equal to the line integral of the particular vector function around that surface.”

What is the divergence of the curl of a vector field?

The divergence of the curl of a vector field is always zero; that means that Even assuming that μ 0 is a constant, that equality simply is not true for all functions. For instance, for f ( x, y) = x and g ( x, y) = 0, it’s false.

Is there a vector field with coordinates 0 in the curl?

Thus F ≡ 0. So, there is not vector field with those coordinates in the curl. Thanks for contributing an answer to Mathematics Stack Exchange! Please be sure to answer the question.

How do you prove a vector field is a conservative vector field?

If →F F → is a conservative vector field then curl →F = →0 curl F → = 0 →. This is a direct result of what it means to be a conservative vector field and the previous fact. If →F F → is defined on all of R3 R 3 whose components have continuous first order partial derivative and curl →F = →0 curl F → = 0 → then →F F → is a conservative vector field.

What is the importance of divergence and curl in calculus?

In this section, we examine two important operations on a vector field: divergence and curl. They are important to the field of calculus for several reasons, including the use of curl and divergence to develop some higher-dimensional versions of the Fundamental Theorem of Calculus.