 everyone myself Piyusha Shedgarh. In this today's lecture we will see the topic Dell operator. These are the learning outcomes. At the end of this session students will be able to define the Dell operator. They will be able to differentiate Dell operator to different coordinate system and they will be able to solve the problems using this Dell operator for the different coordinate system. These are the contents. So, before going to start the Dell operator you can recall which are the different coordinate systems and these coordinate systems are defined with which coordinates that means which coordinates are used for the different coordinate system. Yes, there are the three different coordinate system Cartesian coordinate system which having the coordinates are x, y and z. Second is the cylindrical coordinate system. In cylindrical coordinate system different coordinates r, phi, z are used and the third one is the spherical coordinate system. In spherical coordinate system different coordinates are r, theta, phi and phi are used. Now, define the Dell operator. What is the Dell operator? Dell operator is a differentiation tool that is used in vector differentiation. So, by using Dell operator you can get the differentiation partial differentiation with respect to each axis or each coordinate. It is assumed to have both partial derivative of all coordinate variables as well as the unit vectors. Unit vectors are nothing but the it shows the direction which having the magnitude equal to unity. This is the symbol for the Dell operator. Dell operator is defined in Cartesian coordinate system as this symbol is denoted with the Dell. Dell is dou by dou x a bar x plus dou by dou y a bar y plus dou by dou z a bar z. Whereas, the unit vectors are a bar x a bar y and a bar z. So, the Dell operator is a vector and can be operated on the scalar as well as the vector quantities. It can operate on any vector field for which appropriate partial derivative exist. If the vector field is not changing with respect to space quantities that is x, y and z, then the operation of the Dell on it results in a zero. Now, the Dell operator can be defined in three ways. It can be defined as a gradient, it can be defined as a divergence and also it can be defined as a curl. So, the gradient is nothing but the you are considering with respect to the scalar field. Whereas, the divergence is considered with respect to the vector field and the curl operation is again it is considered with respect to the vector field. With the vector field it can have two operations called the divergence and curl. So, the Dell operator is an operator when applied to a function of one independent variable it yields the derivative. When this function is works on the scalar function it yields the gradient. When it dotted with the any vector field it produces the divergence. Whereas, when this Dell operator crossed with any vector field it produces the curl operation. Divergence of a vector field is nothing but it is a measure of the outgoingness of the field at that point. If you are considering any point then the vector field is diverted in outward direction is nothing but the divergence. If more and more field lines are sourcing out that is coming out of the considered point then whatever is the divergence is the positive divergence. Whereas, when if the field is contracting at a point that means it is incoming at that point sourcing in that case it is considered as the negative divergence. And when the vector field is uniform that is outgoingness equal to the incoming. So, in that case the zero divergence is to be observed. Divergence of a vector v can be written is nothing but the scalar quantity. So, divergence of v is equal to dot operation between this v and the Dell operator. So, it can be written as dou by dou x v x plus dou by dou y v y plus dou by dou z v z. Where Dell operator is nothing but the partial differentiation in cartesian coordinate system with respect to x, y and z. Here the unit vectors are also considered. Whereas, v is the any vector it is defined with the magnitude as well as the direction. Therefore, it can be written in cartesian coordinate system as v x a bar x plus v y a bar y plus v z a bar z. So, this figure shows what is the positive divergence, negative divergence and the zero divergence. So, as you have observed that in first figure this is the first point point p is placed at the center. So, these field lines are going outward direction is going outward direction. So, it is nothing but the positive divergence. Whereas, the point p is placed at the center. In the second figure b, field lines are incoming towards this point p. So, here the divergence is the negative. Whereas, in third figure that is figure number c, outgoing field lines are equal to the incoming field lines and therefore, you are getting the divergence is equal to 0. So, this is the example of the divergence. So, tank is considered the water tank when this water is sourcing out when this water is sourcing out then you can consider that the divergence is positive. That means, the divergence the field lines are going outward. Similar to that the water is pumping in outward direction. So, it is the divergence positive. Here at this point you can consider that this is the outgoing incoming field lines or incoming water at this point and here the same quantity of the water will be leaving through this pipe. So, here the divergence is equal to 0. Whereas, the water is collected in this bucket. So, in this what happened the it is in going in this bucket that means, it is incoming here. It is incoming water is considered here and therefore, the divergence is negative at this point. So, this is the example of the divergence. So, what is the physical significance of the divergence? Physically the divergence of a vector quantity represents the rate of change of the field strength in the direction of the field. So, if the divergence of the field is positive at a point then something is diverging that is outgoing from that point from a small volume surrounding with the point as a source. Whereas, if the it is negative then something is converging into the small volume surrounding that point is acting as a sink. If the divergence of a point is 0 then the rate at which something is entering is equal to the rate at which something is leaving is the same. Therefore, in that case the divergence is 0. So, the vector field whose divergence is 0 is called the solenoidal field. Now, what is the curl of the vector field? The curl of vector field is denoted with this cross product of this vector. So, it can be defined with this determinant a bar x a bar y a bar z dou by dou x dou by dou y dou by dou z v x v y and v z. Where you can see that the del operator is nothing but the partial differentiation with respect to each axis x y and z whereas, the v is any vector v x a bar x plus v y a bar y plus v z a bar z. What is the representation of the curl? So, there are the figures of the representing of the curl. So, in this first figure if you are considering this is the paddle wheel. So, this wheel is not rotating in the first figure there is no any rotation the meaning of this is the curl is equal to 0. Whereas, in second figure the rotation of the paddle wheel which shows the existence of some of the curl is there and this third figure shows the direction of the curl. So, for the vector field whose curl is 0 there is no any rotation of the paddle wheel. In that case these type of the fields are called as a irrotational fields. The gradient of a scalar function phi is a vector whose Cartesian components are dou phi by dou x dou phi by dou y and dou phi by dou z. Then the gradient of phi is given by this equation directly you can multiply with this del operator. So, it becomes dou phi by dou x a bar x plus dou phi by dou y a bar y plus dou phi by dou z a bar z. Different coordinate system which can be defined with the divergence. Divergence can be defined in Cartesian coordinate system as a dou a x by dou x plus dou by dou y a y plus dou by dou z a z. Whereas, the divergence formula can be defined in cylindrical coordinate system as 1 by r dou by dou r r a bar r plus 1 upon r dou a phi by dou phi plus dou a z by dou z. Divergence formula in spherical coordinate system given by this third equation. Now, how to define the gradient? Gradient is defined in different coordinate systems as gradient of v for the Cartesian coordinate system is defined with respect to x, y, z coordinate. Whereas, in cylindrical system it is defined with r phi and z. Again in spherical coordinate system is defined with r theta and phi. How to define the curl in different coordinate system? Now, for that consider any vector field a bar is given by a x a bar x plus a y a bar y plus a z a bar z. So, here the curl is defined in Cartesian coordinate system as curl is nothing but the cross product of any vector field. It is given by this determinant a bar x a bar y a bar z dou by dou x dou by dou y dou by dou z. That means, you are taking the partial differentiation of a x a y and a z. Then the any vector field a is defined with this the cylindrical coordinate system is defined with this curl operation. Again when the vector same vector field is considered so in cylindrical coordinate system the curl operation is defined with this determinant. These are the references for this session.