1. Gradient practice. Compute the gradients of the following functions f in Cartesian, cylindrical, and spherical coordinates. For the non-Cartesian coordinate systems, first use the formula for the gradient in terms of the non-Cartesian unit vectors, and then use the conversions between the unit vectors to convert your answer back to Cartesian coordinates. In all cases, you should find the same answer independent of the coordinate system! (You will find that, for all of these examples, the gradient is easy in one coordinate system but a mess in at least one of the others; this illustrates the value of choosing a good set of coordinates for the problem at hand.) (a) f(x, y, z) = x² + y² + z² (b) f(x, y, z) = sin z (c) f(x, y, z)= x+y+z
1. Gradient practice. Compute the gradients of the following functions f in Cartesian, cylindrical, and spherical coordinates. For the non-Cartesian coordinate systems, first use the formula for the gradient in terms of the non-Cartesian unit vectors, and then use the conversions between the unit vectors to convert your answer back to Cartesian coordinates. In all cases, you should find the same answer independent of the coordinate system! (You will find that, for all of these examples, the gradient is easy in one coordinate system but a mess in at least one of the others; this illustrates the value of choosing a good set of coordinates for the problem at hand.) (a) f(x, y, z) = x² + y² + z² (b) f(x, y, z) = sin z (c) f(x, y, z)= x+y+z
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Transcribed Image Text:1. Gradient practice.
Compute the gradients of the following functions fin
Cartesian, cylindrical, and spherical coordinates. For the non-Cartesian coordinate systems,
first use the formula for the gradient in terms of the non-Cartesian unit vectors, and then use
the conversions between the unit vectors to convert your answer back to Cartesian coordinates.
In all cases, you should find the same answer independent of the coordinate system! (You will
find that, for all of these examples, the gradient is easy in one coordinate system but a mess
in at least one of the others; this illustrates the value of choosing a good set of coordinates
for the problem at hand.)
(a) f(x, y, z) = x² + y² + ₂²
(b) f(x, y, z) = sin z
(c) f(x, y, z)=x+y+z
2. Divergence practice.
Compute the divergences of the following vector fields
v in the given coordinate systems, checking as in problem 1 that you get the same answer
(which should just be a scalar function) in all coordinate systems by converting your final
answer back to Cartesian coordinates.
(a) v = xx+yŷ + zz: Cartesian, cylindrical, spherical
(b) vp: Cartesian, cylindrical
(c) v = 0: Cartesian, spherical
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