Best Answer:
You'd get faster responses if you post things like this in math or physics. I am assuming r is your vector in these is in R3

Let R = xi + yj + zk, where i,j, and k are the unit vectors for each direction.

So unit vector R (which we'll just call r from here on out) is...

r = R/|R| = R/(R.R)^(1/2) = (xi + yj + zk)/(x^2 + y^2 + z^2)^(1/2)

#1

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grad(r) is invalid, since grad is a scalar operator.

div(Er) = E del.r = E ((d/dx)i + (d/dy)j + (d/dz)k).(xi + yj + zk)/(x^2 + y^2 + z^2)^(1/2)

div(r) = E[(y^2 + z^2)/(x^2 + y^2 + z^2)^(3/2)]i + [(x^2 + z^2)/(x^2 + y^2 + z^2)^(3/2)]j + [(x^2 + y^2)/(x^2 + y^2 + z^2)^(3/2)]k

div(r) = E((y^2 + z^2)i + (x^2 + z^2)j + (x^2 + y^2)k)/(x^2 + y^2 + z^2)^(3/2)

curl(r) = del x Er.

Cross products are messy to convey so I will just post the result.

curl(r) = E del x r = 0 vector

#2

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For R.R the only valid operator is actually grad(R.R)

grad(R.R) = d(x^2+y^2+z^2)/dx i + d(x^2+y^2+z^2)/dy j + d(x^2+y^2+z^2)/dz k

grad(R.R) = 2x i + 2y j + 2 z k

grad(R.R) = 2(x i + y j + z k)

grad(R.R) = 2 R

#3

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This one looks like the gravity vector but I am not sure if you meant -GMm/|r| r or what....

If so, then we know r is a unit vector so |r| is just 1. And the answers will be otherwise identical to the first, except instead you will have the constant -GMm instead of E.

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