1. Calculate Vr, wherer=|=x+y+z2: a) Using rectangular (Cartesian) coordinates. b) Using the spherical polar coordinate expression for the gradient operator. 2. Sketch the vector field for the gradient in problem 1. 3. Using rectangular coordinates, prove that for any scalar field f, Vxvf=0. 4. Using rectangular coordinates, prove that for any vector field A, V.(x)=0. 5. Following the technique we used in class to discuss the cylindrical coordinate system, express the differential line element, dx, in terms of the differentials of the spherical polar coordinates, dr, do, and do. 6. a) Express the spherical coordinate system basis vectors, and in terms of the Cartesian coordinate system basis vectors. b) Using your result in part a), express dx in terms of spherical coordinates and spherical coordinate basis vectors. 7. Determine the values of the scale factors in the spherical coordinate system. 8. Write down the position vector, x, of a point in the spherical coordinate system. 9. Using the scale factors you found in Problem 7, determine the differential area element d²x in spherical coordinates. 10.Using the scale factors you found in Problem 7, determine the differential volume element d³x in spherical coordinates. 11 Storing with the coordinate indonandant definition of the radiant of a canlar. 1 /1 10.Using the scale factors you found in Problem 7, determine the differential volume element d³x in spherical coordinates. 11. Staring with the coordinate-independent definition of the gradient of a scalar field (equation 2.38 in Pollack and Stump), and following the method presented in class, derive the expression for the gradient of a scalar field, f, in the spherical coordinate system. Compare your result to the corresponding expression on the inside front cover of your book.