Continuing our discussion of calculus, the last topic I want to discuss here is the concepts of gradient, divergence, and curl. So, first of all we have operators and functions that are of considerable importance in physics and engineering. And the definitions are given in this extract on the right hand side from the manual. And we start off with a Del operator, which is a vector operator, which looks like an upside down triangle, is defined in this way. Partial d by dx i plus d by dyj plus d by dzk in general is the Del operator. So, for example, the gradient of a scalar field, sometimes written as grad of phi, is given here. So, grad phi is equal to d phi by dx times i, etc. And in simple two-dimensional space, this becomes just the local gradient of the field. So the gradient of a scalar field, generally speaking, is a vector quantity. Next, we have the divergence of a vector field. And the divergence of a vector field is defined as the dot product between the Del operator and the vector field itself as written out here. And this has applications, for example, in fluid mechanics. If the vector field V is the fluid velocity, then the divergence of the velocity field becomes the continuity equation. The divergence of the velocity field is equal to 0, which is an equation for conservation of volume in an incompressible fluid. The next property is the curl of a vector field. And the curl of a vector field is defined as the cross product between the Del operator and the vector field. And cross product, therefore, this is a vector quantity itself as defined here. And again, an example from fluid mechanics, if the vector field V is the fluid velocity, then a field which has the curl of the velocity field equal to 0 is called an irrotational flow and it has particular properties. Finally, we have the Laplacian of a scalar field which is Del dot Del times the field, or Del squared of phi, which is given by the equation right here. So, that is d2 phi by dx squared, etc. And, again from fluid mechanics, a particular flow field which has the divergence of the potential function Del squared phi is equal to 0 is called a potential flow. Let's do some examples on that. The divergence of the vector field given by V is equal to 3xi + 5yj is equal to which of these? And, of course, this is a scalar quantity. So, first we do our basic definitions. The divergence is the dot product between the Del operator and the vector field. So, in this case, this becomes d by dx of i, and then write down the vector field here, which is 3xi + 5yi + 0 times k. So, to evaluate this we use our usual rules of evaluating a dot product. In other words, this term multiplied by this term, plus this times this, plus this times this. So, that becomes this. So, we have d by dx of 3x + d by dy of 5y + d by dz of 0. So, evaluating those differentials, that is equal to 3 + 5, which is equal to 8. And so the answer is C. Next, we'll compute the curl of that same vector field 3yi + 5xj. Which of these is it? And again, now we're dealing with a vector quantity cuz curl is a vector quantity computed from the cross product between the Del operator and the vector field. So, this is most easily computed by means of the determinant. So we write this as a determinant with the unit vectors in the top row, the operators in the second row, and finally, the vector field, the components of the vector field, in the third row. And to evaluate this, we can do this by means of the usual rules of evaluating a determinant. So, first of all, the i term here, cross those two terms out. That becomes dvz by dy- dvy by dz, which is this term. Then, the j term, which is negative because the sines alternate with each column. And again, cross out the row and column containing j. So, we have dvz by dx- dvx by dz is this term. And finally, the k component term, cross those two out, is going to be positive k multiplied by dvy by dx- dvx by dy, the last term. So, now we're ready to evaluate this, and writing out the terms, we have this. Vz is 0, so dvz by dy is 0. D times 5x by dz is also 0. And similarly, with the remaining terms, and ultimately when we go through this all we're left with is the k term. And the answer is 2k. So, the answer is C. Final example. What is the Laplacian of the scalar field 2x cubed + xy squared + 2? Which of these alternatives is it? So, start out with our definition of the Laplacian, which is Del squared phi d2 phi by dx squared, etc. So, differentiating this. First, if I differentiate this function once with respect to x, I get 6x squared + y squared. Then, differentiating again, I get d2 phi by dx squared is equal to 12x. Similarly, the second differential of this function with respect to y is equal to 2x. And because there are no z terms in there, the second differential of phi with respect to z is equal to 0. And, adding those up, we get Del squared of phi is equal to 12x + 2x is equal to 14x. And so the answer is B. And this concludes this section