What will be the divergence of a vector field in the spherical coordinate system?

The vector (x, y, z) points in the radial direction in spherical coordinates, which we call the direction. Its divergence is 3. A multiplier which will convert its divergence to 0 must therefore have, by the product theorem, a gradient that is multiplied by itself.

How do you find the divergence of a vector field at a point?

We define the divergence of a vector field at a point, as the net outward flux of per volume as the volume about the point tends to zero. Example 1: Compute the divergence of F(x, y) = 3x2i + 2yj. Solution: The divergence of F(x, y) is given by ∇•F(x, y) which is a dot product.

Does divergence depend on coordinate system?

Divergence is independent of coordinate system by definition.

What is the divergence of a vector?

The divergence of a vector field simply measures how much the flow is expanding at a given point. It does not indicate in which direction the expansion is occuring. Hence (in contrast to the curl of a vector field), the divergence is a scalar.

How do you find the spherical basis of a vector?

The unit vectors in the spherical coordinate system are functions of position. It is convenient to express them in terms of the spherical coordinates and the unit vectors of the rectangular coordinate system which are not themselves functions of position. r = xˆ x + yˆ y + zˆ z r = ˆ x sin! cos” + ˆ y sin!

How do you convert Cartesian to spherical coordinates?

To convert a point from spherical coordinates to Cartesian coordinates, use equations x=ρsinφcosθ,y=ρsinφsinθ, and z=ρcosφ.

Which of the following is an example of vector field?

A gravitational field generated by any massive object is also a vector field. For example, the gravitational field vectors for a spherically symmetric body would all point towards the sphere’s center with the magnitude of the vectors reducing as radial distance from the body increases.

Is curl a vector or scalar?

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.

Why the divergence of curl of any vector is zero?

1 ∇⋅(∇×F)=0. In words, this says that the divergence of the curl is zero. Recalling that gradients are conservative vector fields, this says that the curl of a conservative vector field is the zero vector. Under suitable conditions, it is also true that if the curl of F is 0 then F is conservative.

What is difference between curl and divergence?

The divergence of a vector field is a scalar function. Divergence measures the “outflowing-ness” of a vector field. 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.

How do you convert spherical to Cartesian vector?

To convert a point from spherical coordinates to Cartesian coordinates, use equations x=ρsinφcosθ,y=ρsinφsinθ, and z=ρcosφ. To convert a point from Cartesian coordinates to spherical coordinates, use equations ρ2=x2+y2+z2,tanθ=yx, and φ=arccos(z√x2+y2+z2).

What is the radial unit vector?

The radial unit vector always points one unit in the same direction as the position vector of the point in question (in the direction of increasing radius ). The vector pointing directly away from the origin is the unit vector , and the unit vector at right angles to the radial direction is the transverse unit vector .

What is the divergence of a vector in spherical coordinates?

The vector (x, y, z) points in the radial direction in spherical coordinates, which we call the direction. Its divergence is 3. A multiplier which will convert its divergence to 0 must therefore have, by the product theorem, a gradient that is multiplied by itself.

What is the divergence of a spherical field?

Divergence in Spherical Coordinates Intuitive derivation of the Divergence formula The divergence indicates the outgoingness of the field at the point of interest. Let vector field A is present and within this field say point P is present.

How to find divergence in a coordinate system?

We can find neat expressions for the divergence in these coordinate systems by finding vectors pointing in the directions of these unit vectors that have 0 divergence. Then we write our vector field as a linear combination of these instead of as linear combinations of unit vectors.

Is the divergence of a vector field real?

Note also that the divergence of a vector field is a scalar field: i.e. a real-valued function associated with every point. Now, a vector field can have components in any direction. So, we have in general: