 Hello everyone, myself, Mrs. Mayuri Kangre, Assistant Professor of Mathematics from the Department of Humanities and Sciences, Valchan Institute of Technology, Solapur, today we are going to see vector differentiation. The learning outcome is, at the end of this session, the students will be able to compute the scalar potential function. Let us see the scalar potential function. A scalar potential function is a fundamental concept in a vector analysis and physics. It is a general description of a function used in mathematics and physics to describe the conservative fields. Let us see the definition of scalar potential function. If f bar e is equals to f1i plus f2j plus f3k is an irrotational vector where i, j, k are the unit vectors in the direction of coordinate axis, then there exists a scalar function phi such that f bar e is equals to del phi, then phi is called as scalar potential function. Given that f bar e is equals to del phi where f bar e is an irrotational vector and f bar e is equals to f1i plus f2j plus f3k, del phi is equals to i into dou phi by dou x plus j into dou phi by dou y plus k into dou phi by dou z. We know that when we apply the gradient on scalar potential function phi, we get grad phi which is denoted by a vector f bar. We know the procedure of obtaining this gradient of function phi. Now here in the reverse way, we have the f bar which has the relation grad phi and f bar e is an irrotational vector. In such cases, this scalar potential function phi is obtained from the vector f bar as f bar e is equals to del phi, we can write it as f1i plus f2j plus f3k is equals to i into dou phi by dou x plus j into dou phi by dou y plus k into dou phi by dou z. Here we will equate the coefficients of i, j, k from both the sides which gives us f1 e is equals to dou phi by dou x, f2 equals to dou phi by dou y and f3 equals to dou phi by dou z. Now we know the definition of total derivative using it, we can write d phi is equals to dou phi by dou x into dx plus dou phi by dou y into dy plus dou phi by dou z into dz. Here we will replace dou phi by dou x, dou phi by dou y and dou phi by dou z with f1, f2 and f3 respectively which gives us d phi is equals to f1 dx plus f2 dy plus f3 dz. On integration, we get the function phi equals to integration of f1 dx treating y z as constant plus integration of terms of f2 free from x with respect to y treating z as constant plus the integration of terms of f3 free from x and y with respect to z with this formula we can obtain the scalar potential function phi. Now we will see the procedure to find the scalar potential function given that f bar is equals to del phi where f bar is an irrotational vector with f bar equals to f1i plus f2j plus f3k. Step one, we will find out f1 f2 f3 which is given the given vector f bar and we will write down the equation f bar equals to del phi by f1i plus f2j plus f3k equals to i into dou phi by dou x plus j into dou phi by dou y plus k into dou phi by dou z. Step two, write d phi in terms of f1 f2 f3 that is we can write d phi equals to f1 dx plus f2 dy plus f3 dz. Now step three, we will integrate this equation with the formula phi is equals to integration of f1 dx treating y z as constant plus integration of terms of f2 free from x with respect to y treating z as constant plus integration of terms of f3 free from x and y with respect to z. Now let us go for examples. First example, show that the vector f bar is equals to x square minus y z into i plus y square minus zx into j plus z square minus xy into k is an irrotational and also find its scalar potential function. Before going to find out the scalar potential function, first we have to show that the vector f bar is an irrotational vector. I hope everyone knows the meaning of irrotational vector. Now pause the video for a minute and give the answer of this question. When we say that a vector is irrotational, I hope everyone knows the answer. We will check the answer. When the curl of f bar is a zero vector that is a del cross f bar is equal to zero vector. Then we say that the vector f bar is an irrotational vector that is the determinant with the first row entries i, j, k, second row entries doh by doh x, doh by doh y, doh by doh z and the third row entries f1, f2, f3 if this determinant value is a zero vector then we say that the vector f bar is an irrotational vector. Now f bar is given as x square minus yz into i plus y square minus zx into j plus z square minus xy into k gives us f1 equals to x square minus yz, f2 equals to y square minus zx and f3 equals to z square minus xy. We will substitute these f1, f2, f3 in the definition of curl of f bar that is del cross f bar equals to we get this determinant and now we will solve the determinant. So we can get del cross f bar equals to i into bracket doh by doh y of z square minus xy minus doh by doh z of y square minus zx minus j into doh by doh x of z square minus xy minus doh by doh z of x square minus yz plus k into doh by doh x of y square minus zx minus doh by doh y of x square minus yz. Now we will simplify it. Let us see the first term i into bracket. The first term is doh by doh y of z square minus xy. Here the differentiation is with respect to y treating x and z as constant. So the derivative of z square will be zero and the derivative of minus xy with respect to y is minus x. Similarly, in the second term the differentiation is with respect to z. So the derivative of y square will be zero and for this minus zx the derivative will be minus x. So we get i into bracket minus x minus of minus x. Similarly, minus j into bracket we get minus y minus of minus y plus k into bracket minus z minus of minus z. Now we know that the minus minus becomes plus. So we get i into bracket minus x plus x minus j into bracket minus y plus y plus k into bracket minus z plus z which gives us del cross f bar equals to zero i minus zero j plus zero k that is zero vector. So f bar is an irrotational vector. Now we will go for obtaining the scalar potential function. To find this scalar potential function first we have to show that the f bar is an irrotational vector which we have already proved. Now let us find out the scalar potential function we will call it as phi such that f bar is equals to del phi that is f1i plus f2j plus f3k is equals to i into doh phi by doh x plus j into doh phi by doh y plus k into doh phi by doh z. As d phi is equals to doh phi by doh x into dx plus doh phi by doh y into dy plus doh phi by doh z into dz replacing these partial derivatives with f1f2f3 we can write d phi is equals to f1dx plus f2dy plus f3dz. We know that f1 is x square minus yz, f2 is y square minus zx and f3 z square minus xy. So we can write d phi is equals to x square minus yz into dx plus y square minus zx into dy plus z square minus xy into dz. On integration we get phi with this formula which we have already seen. Now in this formula we will replace f1f2 and f3. f1 is x square minus yz, f2 is y square minus zx and f3 is z square minus xy. So phi is equals to integration of x square minus yz into dx treating yz constant plus for the term f2 we have a y square minus zx and we have to take the terms of f2 free from x. So we will omit the term minus zx so we will consider only y square. So the second term will be integration of y square dy treating z as constant plus from f3 we are going to consider the terms which are free from x and y. So we will omit this minus xy and we will consider only z square from this f3 so integration of z square dz. So we get d phi is equals to the integration of x square with respect to x is x cube by 3 minus sign as it is yz is treated as constant. So the integration of yz with respect to x will be xyz plus the integration of y square is y cube by 3 plus the integration of z square is z cube by 3. So rewriting phi we get phi equals to taking 1 by 3 common into the bracket x cube plus y cube plus z cube minus 3xyz. Thank you.