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Calculate $R$ in $\mathrm{L} \cdot \mathrm{atm} / \mathrm{mol} \cdot \mathrm{K},$ in $\mathrm{kPa} \cdot \mathrm{dm}^{3} / \mathrm{mol} \cdot \mathrm{K},$ in $\mathrm{J} / \mathrm{mol} \cdot \mathrm{K},$ and in $\mathrm{kJ} / \mathrm{mol} \cdot \mathrm{K}$

We were given a set of fields and were asked Thio use Stokes is theorem to show that these circulations of these fields around the boundary of any smooth oriental surface in space are zero. So in part a, the old F is two x i plus two y j plus two zk Well, it follows that the curl of us is going to be zero and therefore we have that the circulation of this field around the boundary of any smooth oriental surface See, the boundary is C Well, by Stokes is the're, um this is equal to the double integral over the surface s of the curl of f in the direction of the outward normal vector n. But as we preview previously calculated, this is zero and therefore our whole integral is zero. In part being were given the field f equals the ingredients of X y squared z cute so well, Let little left of X y z We are scaler function x y squared Is he cute? And it follows that the curl of f bigger. This is equal to the ingredient crossed with the Grady int of little laugh and we know by equation eight from this section. But this is equal to zero vector, the radiant crossed with ingredient. The function is zero and therefore we have that the curl of death is zero. So ex Stokes is the're, um If C is the boundary of any smooth oriental surface in space in the circulation of this field, oversee is equal to the flux of the curl of F Across the surface s in the direction of the outward unit Normal end. But as you pointed out before the curl of F zero, So this is integral of zero, which of course, is zero in part C were given the field the ingredient crossed with X I plus yj plus zk Well, in fact, we see that our function f so this is the curl of the vector function x i plus yj plus ck which we see right away zero and therefore we knew that the curl of the zero function is also zero. And so we have that if c is the boundary of any smooth oriental surface s in space. The circulation of this field around see by Stokes theorem is equal to the flux of the curl of Yes, cross s in the direction of the outward unit Normal. And well, as you pointed out, the curl of F zero. So this is equal to the integral of zero, which of course, is zero. And finally, in part D were given that f is a conservative vector field or radiant vector field ingredients of little left. So from this, it follows that he curl of Big Gap. This is the curl of the Grady int little left and again from Equation eight from the section The curl of the Grady int of a function is always zero zero vector and therefore by Stokes is the're, um If C is the boundary of any smooth oriental surface s in space, we have that the circulation of big F around see is equal to the flux of the curl of big F Across s in the direction of the Outward Unit Normal vector. And but again, he already calculated the curl with zero. So this is the integral of zero, which is zero. And so you've shown for all these function

Here I can write develop uh is equal to p m dot this multiplication B. So this will tend to f minus B. Integration zey dot this B d p now have physical too. This bye immune, not immune, mu mu not integration of be dot to this multiplication B Dp since be predominant mentally along the X axis. So I can write the value of app X required to this by um you may not integration B X this by this sbx which is equal to this as By 2 μ μ not μ integration of X is equal to 02. X is equal dwell D B. Act as glad which is equal to minus this as B squared by two mu mu not which is approximately equal to this as b squared by two mu not as the answer.