Divergence and Curl of a Vector Field Electrical Engineering (EE) Notes | EduRev

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Electrical Engineering (EE) : Divergence and Curl of a Vector Field Electrical Engineering (EE) Notes | EduRev

The document Divergence and Curl of a Vector Field Electrical Engineering (EE) Notes | EduRev is a part of the Electrical Engineering (EE) Course Electromagnetic Theory.
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we had defined divergence of a vector field and had obtained an expression for the divergence in the Cartesian coordinates. We also derived the divergence theorem which connects the flux of a vector field with the volume integral of the divergence of the field.

Physically, the divergence, as the name suggests is a measure of the amount of spread that the field has at a point.

Divergence and Curl of a Vector Field Electrical Engineering (EE) Notes | EduRevDivergence and Curl of a Vector Field Electrical Engineering (EE) Notes | EduRev

For instance, in the figure above, the vector field shown to the left has a positive divergence while that to the right has a negative divergence. In electrostatics, we will see that the field produced by a positive charge has positive divergence while a negative charge produces an electrostatic field with negative divergence.

Divergence, curl etc. were extensively used in fluid dynamics from which a lot of nomenclature have arisen. Let us consider a fluid flowing through an elemental volume of dimension Divergence and Curl of a Vector Field Electrical Engineering (EE) Notes | EduRev with its sides oriented parallel to the Cartesian axes. In the figure below, we show only the y-component of the velocity of the fluid entering and leaving the elemental volume. Let the density of the fluid at Divergence and Curl of a Vector Field Electrical Engineering (EE) Notes | EduRev and the y-component of the velocity be vy.  We define a vector  Divergence and Curl of a Vector Field Electrical Engineering (EE) Notes | EduRev The mass of the fluid flowing into the volume per unit time through the left face which has an outward normal  Divergence and Curl of a Vector Field Electrical Engineering (EE) Notes | EduRev is given by

Divergence and Curl of a Vector Field Electrical Engineering (EE) Notes | EduRev

Divergence and Curl of a Vector Field Electrical Engineering (EE) Notes | EduRev

Mass of the fluid flowing out is  Divergence and Curl of a Vector Field Electrical Engineering (EE) Notes | EduRev Retaining only the first order term in a Taylor series expansion, we have

Divergence and Curl of a Vector Field Electrical Engineering (EE) Notes | EduRev

Thus the net increase in mass is  Divergence and Curl of a Vector Field Electrical Engineering (EE) Notes | EduRev This is the increase due to the y-component of the velocity. We can write similar expressions for the flow in the x and z directions. The net increase in mass per unit time is Divergence and Curl of a Vector Field Electrical Engineering (EE) Notes | EduRev where we have put the volume element Divergence and Curl of a Vector Field Electrical Engineering (EE) Notes | EduRev so as not to confuse with the vector  Divergence and Curl of a Vector Field Electrical Engineering (EE) Notes | EduRev defined above.

Another of of talking about the net increase in mass is to realize that since the volume is fixed, the increase in mass is due to a change in the density alone. Thus the rate of increase of mass is Divergence and Curl of a Vector Field Electrical Engineering (EE) Notes | EduRev

Equating these two expressions, we get what is known as the equation of continuity in fluid dynamics,

Divergence and Curl of a Vector Field Electrical Engineering (EE) Notes | EduRev

To see what this equation implies, consider, for example a vector field given by  Divergence and Curl of a Vector Field Electrical Engineering (EE) Notes | EduRev The field has been plotted using Mathematica (see figure)

Divergence and Curl of a Vector Field Electrical Engineering (EE) Notes | EduRevDivergence and Curl of a Vector Field Electrical Engineering (EE) Notes | EduRev

To the left is plotted  Divergence and Curl of a Vector Field Electrical Engineering (EE) Notes | EduRev Note that in the first and the third quadrants the divergence is positive while in the other two quadrants it is negative. The figure to the right is for the force field  Divergence and Curl of a Vector Field Electrical Engineering (EE) Notes | EduRev which has zero divergence.

The size of the arrow roughly represents the magnitude of the vector. The divergence is given by 4xy. In the first and the third quadrants (x,y both positive or both negative) divergence is positive. One can see that in these quadrants, if you take any closed region the size of the arrows which are entering the region are smaller than those leaving it. Thus the density decreases, divergence is positive. Reverse is true in the even quadrants.

In the figure to the right, the force field Divergence and Curl of a Vector Field Electrical Engineering (EE) Notes | EduRev has zero divergence. If you take a closed region in this figure, you find as many vectors are getting in as are going out. The field is a solenoidal (zero divergence) field.

Divergence Theorem – Examples :

Recall divergence theorem  Divergence and Curl of a Vector Field Electrical Engineering (EE) Notes | EduRev where the surface integral is taken over a closed surface defining the enclosed volume. As an example consider the surface integral of the position Divergence and Curl of a Vector Field Electrical Engineering (EE) Notes | EduRev over the surface of a cylinder of radius a and height h.

Evaluating the surface integral by use of the divergence theorem is fairly simple. Divergence of position vector has a value 3 because  Divergence and Curl of a Vector Field Electrical Engineering (EE) Notes | EduRev Thus the volum integral of the divergence is simply three times the volume of the cylinder which gives Divergence and Curl of a Vector Field Electrical Engineering (EE) Notes | EduRev

Direct calculation of the surface integral can be done as follows. For convenience, let the base of the cylinder be in the x-y plane with its centre at the origin.

Divergence and Curl of a Vector Field Electrical Engineering (EE) Notes | EduRev

There are three surface of the cylinder, a top cap, a bottom cap and the curved surface. For the top cap, the normal vector  Divergence and Curl of a Vector Field Electrical Engineering (EE) Notes | EduRev direction, so that Divergence and Curl of a Vector Field Electrical Engineering (EE) Notes | EduRev On the top surface z is constant and is given by z=h. Thus  Divergence and Curl of a Vector Field Electrical Engineering (EE) Notes | EduRev For the bottom cap Divergence and Curl of a Vector Field Electrical Engineering (EE) Notes | EduRev direction so that Divergence and Curl of a Vector Field Electrical Engineering (EE) Notes | EduRev However, the value of z on this surface being zero, the flux vanishes. We are now left with the curved surface for which the outward normal is parallel to the x-y plane. The unit vector on this surface is Divergence and Curl of a Vector Field Electrical Engineering (EE) Notes | EduRev However, on the surface Divergence and Curl of a Vector Field Electrical Engineering (EE) Notes | EduRevDivergence and Curl of a Vector Field Electrical Engineering (EE) Notes | EduRev Thus the surface integral is Divergence and Curl of a Vector Field Electrical Engineering (EE) Notes | EduRev Adding to this the contribution from the top and the bottom face, the surface integral works out to Divergence and Curl of a Vector Field Electrical Engineering (EE) Notes | EduRev as was obtained from the divergence theorem

As a second example, consider a rather nasty looking vector field Divergence and Curl of a Vector Field Electrical Engineering (EE) Notes | EduRevDivergence and Curl of a Vector Field Electrical Engineering (EE) Notes | EduRev over the surface of a cubical box  Divergence and Curl of a Vector Field Electrical Engineering (EE) Notes | EduRev We will not attempt to calculate the surface integral directly. However, the divergence theorem gives a helping hand.

Divergence and Curl of a Vector Field Electrical Engineering (EE) Notes | EduRev

We need to calculate the triple integral  Divergence and Curl of a Vector Field Electrical Engineering (EE) Notes | EduRev As the integrand has no z dependence, the z-integral evaluates to 1. The volume integral is Divergence and Curl of a Vector Field Electrical Engineering (EE) Notes | EduRev

Curl of a Vector Field :

We have seen that the divergence of a vector field is a scalar field. For vector fields it is possible to define an operator which acting on a vector field yields another vector field. The name curl comes from “circulation” which measures how much does a vector field “curls” about a point.

Consider an open surface of the type shown – something like an inverted pot with a rim. We wish to calculate the surface integral of a vector field defined over this surface.
Divergence and Curl of a Vector Field Electrical Engineering (EE) Notes | EduRev

Let us divide the surface into a large number of small segments and calculate the line integral over the boundary of such an elemental surface. One such elemental surface with boundary is shown. If we define the bounding curve of this surface in the counterclockwise direction, the normal to such a surface will be outward. If we consider each such segment i and take the line integral over the bounding curve ΔCi the contribution to the line integral from adjacent regions will cancel because the line integral from the boundary are traversed in opposite sense (look at the black and the red arrows on the boundary of two segments). Considering all such segments, we will be only left with the uncompensated line integral at the rim. This is illustrated in the following figure :

The line integral on the boundary for ABCD + line integral for The boundary of CDEF is equal to the line integral for ABEF because the integral is traversed along CD in the former and along DC in the latter.

Divergence and Curl of a Vector Field Electrical Engineering (EE) Notes | EduRev

Thus we have,

Divergence and Curl of a Vector Field Electrical Engineering (EE) Notes | EduRev    (1)

The quantity in the parenthesis in the last expression is defined as the curl of the vector field  Divergence and Curl of a Vector Field Electrical Engineering (EE) Notes | EduRev limit of the elemental surface goes to zero. Curl being a vector, its direction is specified as the outgoing normal to the surface element.

Divergence and Curl of a Vector Field Electrical Engineering (EE) Notes | EduRev

It may be noted that because the definition is valid in the limit of the surface area going to zero, it is a point relationship. Using this definition, we can write the previous equation (1) as

Divergence and Curl of a Vector Field Electrical Engineering (EE) Notes | EduRev

This equation relates the surface integral of the curl of a vector field with the line integral of the vector field and is known as “Stoke’s Theorem”.

Tutorial

1. Calculate the flux of the vector field  Divergence and Curl of a Vector Field Electrical Engineering (EE) Notes | EduRev over the surface of a unit cube whose edges are parallel to the axes and one of the corners is at the origin. Use this result to illustrate the divergence theorem.

2. Calculate the flux of the vector field Divergence and Curl of a Vector Field Electrical Engineering (EE) Notes | EduRev over the surface of a unit sphere. Use this result to illustrate the divergence theorem. (Use spherical coordinates).

3. Calculate the flux of the vector Divergence and Curl of a Vector Field Electrical Engineering (EE) Notes | EduRev over the surface of a right circular cylinder of radius R bounded by the surfaces z=0 and z=h. Calculate it directly as well as by use of the divergence theorem.

4. Calculate the flux of the position vector Divergence and Curl of a Vector Field Electrical Engineering (EE) Notes | EduRev through a torus of inner radius a and outer radius b. Use the result to illustrate divergence theorem. (* This is a hard problem).

5. Calculate the flux of the vector field  Divergence and Curl of a Vector Field Electrical Engineering (EE) Notes | EduRev over the surface defined by  Divergence and Curl of a Vector Field Electrical Engineering (EE) Notes | EduRevDivergence and Curl of a Vector Field Electrical Engineering (EE) Notes | EduRev

Solution to Tutorial Problems :

1. The geometry of the cube along with the direction of surface normalsare shown in the figure. Consider the base of the cube which is the plane z=0. On this face  Divergence and Curl of a Vector Field Electrical Engineering (EE) Notes | EduRev Since the normal is along the -z direction flux from this face is zero. Similarly, the flux from the other two faces which meet at the origin are also zero. Consider the top face where z=1. On this face Divergence and Curl of a Vector Field Electrical Engineering (EE) Notes | EduRevDivergence and Curl of a Vector Field Electrical Engineering (EE) Notes | EduRev The normal is in the +z direction, so that the flux is  Divergence and Curl of a Vector Field Electrical Engineering (EE) Notes | EduRev Likewise, the flux from the other two faces are also ½ each. The total flux, therefore, is 3/2. The divergence of the field Divergence and Curl of a Vector Field Electrical Engineering (EE) Notes | EduRev

Divergence and Curl of a Vector Field Electrical Engineering (EE) Notes | EduRev

Divergence and Curl of a Vector Field Electrical Engineering (EE) Notes | EduRev

The volume integral of the divergence is Divergence and Curl of a Vector Field Electrical Engineering (EE) Notes | EduRev

By symmetry, this is 3 times  Divergence and Curl of a Vector Field Electrical Engineering (EE) Notes | EduRev 1/2. Thus the volume integral of the divergence is 3/2.

2. Divergence of the field is Divergence and Curl of a Vector Field Electrical Engineering (EE) Notes | EduRev Thus the volume integral of the divergence over the surface of a unit sphere is just   Divergence and Curl of a Vector Field Electrical Engineering (EE) Notes | EduRev To calculate the surface integral we note that the normal on the surface of the sphere is along the radial direction and is given by Divergence and Curl of a Vector Field Electrical Engineering (EE) Notes | EduRev where R=1 is the radius of the sphere. Thus Divergence and Curl of a Vector Field Electrical Engineering (EE) Notes | EduRev Since the surface element on the sphere is Divergence and Curl of a Vector Field Electrical Engineering (EE) Notes | EduRev we have , substituting (in spherical polar) Divergence and Curl of a Vector Field Electrical Engineering (EE) Notes | EduRevDivergence and Curl of a Vector Field Electrical Engineering (EE) Notes | EduRev
   Divergence and Curl of a Vector Field Electrical Engineering (EE) Notes | EduRev

The first term gives zero because of vanishing of the integral over φ. We are left with Divergence and Curl of a Vector Field Electrical Engineering (EE) Notes | EduRev

3. Let the base of the cylinder be at z=0 and the top at z=h. The origin is at the centre of the base. The cylinder has three surfaces. For the bottom surface, the direction of the normal is along  Divergence and Curl of a Vector Field Electrical Engineering (EE) Notes | EduRev and on this surface z=0. The surface integral for this surface is  Divergence and Curl of a Vector Field Electrical Engineering (EE) Notes | EduRev For the top surface, the normal is along Divergence and Curl of a Vector Field Electrical Engineering (EE) Notes | EduRev the surface integral is Divergence and Curl of a Vector Field Electrical Engineering (EE) Notes | EduRev For the curved surface the direction of the normal is outward radial direction in the x-y plane which is Divergence and Curl of a Vector Field Electrical Engineering (EE) Notes | EduRev so that the surface integral is Divergence and Curl of a Vector Field Electrical Engineering (EE) Notes | EduRev The integral is done in the cylindrical coordinates by polar substitution Divergence and Curl of a Vector Field Electrical Engineering (EE) Notes | EduRev The surface element is Divergence and Curl of a Vector Field Electrical Engineering (EE) Notes | EduRev Thus the integral become The angle integral in both cases gives zero. Thus the total flux is only contributed by the top surface and is  Divergence and Curl of a Vector Field Electrical Engineering (EE) Notes | EduRev The angle integral in both cases gives zero. Thus the total flux is only contributed by the top surface and is Divergence and Curl of a Vector Field Electrical Engineering (EE) Notes | EduRev This can also be seen by the divergence theorem. Divergence and Curl of a Vector Field Electrical Engineering (EE) Notes | EduRev The volume integral is Divergence and Curl of a Vector Field Electrical Engineering (EE) Notes | EduRev

4. Geometrically a torus is obtained by taking a circle, say in the x-z plane and rotating it about the z-axis to obtain a solid of revolution. Let us define the mean radius of the torus to be  Divergence and Curl of a Vector Field Electrical Engineering (EE) Notes | EduRev and the radius of the circle which is being revolved about the z-axis to be Divergence and Curl of a Vector Field Electrical Engineering (EE) Notes | EduRev The position vector of an arbitrary point on the torus is defined as follows.

Divergence and Curl of a Vector Field Electrical Engineering (EE) Notes | EduRev

Consider the coordinate of an arbitrary point on the circle which is in the x-z plane. Let the position of the point make an angle φ with the x-axis. The coordinate of this point is  Divergence and Curl of a Vector Field Electrical Engineering (EE) Notes | EduRev When the circle is rotated about the z-axis by an angle θ, the z coordinate does not change. However, the x and y coordinates change and become
Divergence and Curl of a Vector Field Electrical Engineering (EE) Notes | EduRev
Thus an arbitrary point on the torus can be parameterized by  Divergence and Curl of a Vector Field Electrical Engineering (EE) Notes | EduRev given by the above expressions. A surface element on the torus is then obtained by the area formed by an arc obtained by incrementing Divergence and Curl of a Vector Field Electrical Engineering (EE) Notes | EduRev and the arc formed by incrementing  Divergence and Curl of a Vector Field Electrical Engineering (EE) Notes | EduRev The area element is therefore given by the cross product Divergence and Curl of a Vector Field Electrical Engineering (EE) Notes | EduRev (The unit vector on the surface is directed along the direction of the cross product.). The partial derivatives are given by
Divergence and Curl of a Vector Field Electrical Engineering (EE) Notes | EduRev

Thus

Divergence and Curl of a Vector Field Electrical Engineering (EE) Notes | EduRevDivergence and Curl of a Vector Field Electrical Engineering (EE) Notes | EduRev (the order of the cross product determines the outward normal). Thus , substituting Divergence and Curl of a Vector Field Electrical Engineering (EE) Notes | EduRevDivergence and Curl of a Vector Field Electrical Engineering (EE) Notes | EduRev

Divergence and Curl of a Vector Field Electrical Engineering (EE) Notes | EduRev
Divergence and Curl of a Vector Field Electrical Engineering (EE) Notes | EduRev
Divergence and Curl of a Vector Field Electrical Engineering (EE) Notes | EduRev

Since the integrand is independent of θ the integral over it gives 2π. The integral over the remaining angle is straightforward. We can simplify the integrand as follows :

Divergence and Curl of a Vector Field Electrical Engineering (EE) Notes | EduRev

The second and the third term in the integral vanish, the remaining two terms give Divergence and Curl of a Vector Field Electrical Engineering (EE) Notes | EduRev which makes the total contribution to the surface integral as Divergence and Curl of a Vector Field Electrical Engineering (EE) Notes | EduRev
However, the problem is straightforward if we apply the divergence theorem. The divergence of the position vector is 3. Thus by divergence theorem, the surface integral is 3 times the volume of the toroid. The volume of the toroid is rather easy to calculate if we note that if we cut it along a section, the toroid becomes a cylinder of radius r and length 2πR. Thus the volume of the toroid is  Divergence and Curl of a Vector Field Electrical Engineering (EE) Notes | EduRev Thus 3 times the volume is Divergence and Curl of a Vector Field Electrical Engineering (EE) Notes | EduRev consistent with our direct evaluation of the surface integral.

5. The surface is sketched below. Since  Divergence and Curl of a Vector Field Electrical Engineering (EE) Notes | EduRev the region of interest is Divergence and Curl of a Vector Field Electrical Engineering (EE) Notes | EduRev Notice that the divergence theorem is not directly applicable because the surface is not closed.
However, one can close the surface by adding a cap to the surface at z=3. We will calculate the flux by applying the divergence theorem to this closed surface and then explicitly subtract the surface integral over the cap.

The divergence of the field is given by  Divergence and Curl of a Vector Field Electrical Engineering (EE) Notes | EduRev

Thus the surface integral over the closed surface is thus given by  Divergence and Curl of a Vector Field Electrical Engineering (EE) Notes | EduRev To evaluate this, consider a disk lying between z and z+dz. The circular disk of width dz has a volume Divergence and Curl of a Vector Field Electrical Engineering (EE) Notes | EduRev This gives the surface integral to be 4π. We have to now subtract from this the surface integral over the cap that was added by us, which is directed along Divergence and Curl of a Vector Field Electrical Engineering (EE) Notes | EduRev direction. On this surface z=3 so that the radius of the disk is 1. This integral Divergence and Curl of a Vector Field Electrical Engineering (EE) Notes | EduRev can be easily calculated and shown to have value zero. Thus the required surface integral has a value 4π.

Self Assessment Quiz

In the following questions calculate the flux both by direct integration and also by application of the divergence theorem.

1. Calculate the flux of the field Divergence and Curl of a Vector Field Electrical Engineering (EE) Notes | EduRev over the surface of a right circular cylinder of radius R and height h in the first octant, i.e. in the region (x>0, y>0, z>0).
2. Evaluate the surface integral of the vector field  Divergence and Curl of a Vector Field Electrical Engineering (EE) Notes | EduRev over the surface of a unit cube with the origin being at one of the corners.
3. Calculate the flux of  Divergence and Curl of a Vector Field Electrical Engineering (EE) Notes | EduRev over the surface of a sphere of radius R with its centre at the origin.
4. Calculate the flux of Divergence and Curl of a Vector Field Electrical Engineering (EE) Notes | EduRev through the surface defined by a cone Divergence and Curl of a Vector Field Electrical Engineering (EE) Notes | EduRev

5. Evaluate the flux through an open cone  Divergence and Curl of a Vector Field Electrical Engineering (EE) Notes | EduRev for the field Divergence and Curl of a Vector Field Electrical Engineering (EE) Notes | EduRev

Solutions toSelf Assessment Quiz

1. For the curved surface of the cylinder, the unit vector is  Divergence and Curl of a Vector Field Electrical Engineering (EE) Notes | EduRev which gives Divergence and Curl of a Vector Field Electrical Engineering (EE) Notes | EduRev Parameterize  Divergence and Curl of a Vector Field Electrical Engineering (EE) Notes | EduRev Since we are confined to the first octant Divergence and Curl of a Vector Field Electrical Engineering (EE) Notes | EduRev The flux through the slant surface is  Divergence and Curl of a Vector Field Electrical Engineering (EE) Notes | EduRev The top and the bottom caps are in the Divergence and Curl of a Vector Field Electrical Engineering (EE) Notes | EduRev the contribution from these two give zero by symmetry. There are two more surfaces if we consider the first octant, they are the positive x-z plane and the positive y-z plane., the normal to the former being in the direction of  Divergence and Curl of a Vector Field Electrical Engineering (EE) Notes | EduRev that for the latter is along Divergence and Curl of a Vector Field Electrical Engineering (EE) Notes | EduRev The flux from the former is Divergence and Curl of a Vector Field Electrical Engineering (EE) Notes | EduRev while that from the latter is  Divergence and Curl of a Vector Field Electrical Engineering (EE) Notes | EduRevAdding up all the contributions, the total flux from the closed surface is zero. This is consistent with the fact that the divergence of the field is zero

2. There are six faces. For the face at x=0, since the surface is directed along  Divergence and Curl of a Vector Field Electrical Engineering (EE) Notes | EduRev the surface integral is Divergence and Curl of a Vector Field Electrical Engineering (EE) Notes | EduRev The face at x=1 gives +1/2. The faces at y=0 and that at z=0 gives zero because the field is proportional to y and z respectively. The contribution to flux from y=1 is 1 and that from z=1 is 3/2. Adding, the flux is 5/2 units. This can also be done by the divergence theorem. Divergence of the field is 2x+3y, so that the volume integral is Divergence and Curl of a Vector Field Electrical Engineering (EE) Notes | EduRev

3. The divergence of the given vector field is Divergence and Curl of a Vector Field Electrical Engineering (EE) Notes | EduRev Thus, by divergence theorem, the flux is Divergence and Curl of a Vector Field Electrical Engineering (EE) Notes | EduRev We can show this result by direct integration. The unit normal on the surface of the sphere is given by Divergence and Curl of a Vector Field Electrical Engineering (EE) Notes | EduRev that the flux is Divergence and Curl of a Vector Field Electrical Engineering (EE) Notes | EduRev This integral can be conveniently evaluated in a spherical polar coordinates with  Divergence and Curl of a Vector Field Electrical Engineering (EE) Notes | EduRev The surface element on the sphere is Divergence and Curl of a Vector Field Electrical Engineering (EE) Notes | EduRev By symmetry, the flux can be seen to be Divergence and Curl of a Vector Field Electrical Engineering (EE) Notes | EduRev. (Note that we decided to do the integral involving z4 rather than x4 or y4 because the azimuthal integral gives 2π in this case. The integral is easy to perform with the substitution Divergence and Curl of a Vector Field Electrical Engineering (EE) Notes | EduRev which gives the flux to be Divergence and Curl of a Vector Field Electrical Engineering (EE) Notes | EduRev

4. The divergence of the field is 3. The flux, therefore, is 3 times the volume of the cone which is Divergence and Curl of a Vector Field Electrical Engineering (EE) Notes | EduRev The flux is thus π. The direct calculation of the flux involves two surfaces, the slant surface and the cap, as shown in the figure. The cap is in the xy plane and has an outward normal Divergence and Curl of a Vector Field Electrical Engineering (EE) Notes | EduRev (because on the cap z=1 and the cap is a disk of unit radius). Thus it remains to be shown that the flux from the slanted surface vanishes. At any height z, the section parallel to the cap is a circle of radius z. Since, the height and the radius of the cap are 1 each, the semi angle of the cone is 450.

Divergence and Curl of a Vector Field Electrical Engineering (EE) Notes | EduRev

Thus the normal to the slanted surface has a component Divergence and Curl of a Vector Field Electrical Engineering (EE) Notes | EduRev along the z direction and  Divergence and Curl of a Vector Field Electrical Engineering (EE) Notes | EduRev in the x-y plane. The component in the xy plane can be parameterized by the azimuthal angle φ and we can write  Divergence and Curl of a Vector Field Electrical Engineering (EE) Notes | EduRev The area element can be written as Divergence and Curl of a Vector Field Electrical Engineering (EE) Notes | EduRev appears because the length element is along the slant. Thus the contribution from the slanted surface is  Divergence and Curl of a Vector Field Electrical Engineering (EE) Notes | EduRevDivergence and Curl of a Vector Field Electrical Engineering (EE) Notes | EduRev this integral can be evaluated and shown to be zero.

5. This problem is to be attempted similar to the problem 5 of the tutorial, i.e., by closing the cap and subtracting the contribution due to the cap. The divergence being 3, the flux from the closed cone is 3 times the volume of the cone which gives 8π. The contribution from the top face (which is a disk of radius 2) is  Divergence and Curl of a Vector Field Electrical Engineering (EE) Notes | EduRev the net flux is zero. (You can also try to get this result directly as done in problem 4, where we showed that the flux from the curved surface is zero).

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