answer:The whole number 0 is equal to the decimal number 0.0 and the mixed number 0 frac{0}{10}. This is because 0.0 can be written as frac{0}{10}, and frac{0}{10} is equal to 0 because any number divided by itself is equal to 1, and 0 times 1 = 0. Therefore, 0 = 0.0 = 0 frac{0}{10}.

answer:The factored form of the given quadratic expression is 3left(x - sqrt{5}right)left(x + 6sqrt{5}right).

answer:The definition of convergence in a metric space, as given in "Baby Rudin", requires that there exists a point p in the space such that the distance between x_n and p approaches zero as n approaches infinity. In the case of the sequence x_n = frac{1}{n}, it is clear that this sequence converges to zero in the real numbers, since for any epsilon > 0, we can choose N such that n > N implies left| x_n - 0 right| = frac{1}{n} < epsilon. However, in the positive real numbers, which is a subset of the real numbers, the point 0 is not included. Therefore, according to the definition of convergence in a metric space, the sequence x_n = frac{1}{n} does not converge to zero in the positive real numbers, since there is no point p in the positive real numbers such that the distance between x_n and p approaches zero as n approaches infinity.

answer:The curl of a vector field mathbf{F} = fmathbf{hat{i}} + gmathbf{hat{j}} + hmathbf{hat{k}} is given by: nabla times mathbf{F} = left(frac{partial h}{partial y} - frac{partial g}{partial z}right)mathbf{hat{i}} + left(frac{partial f}{partial z} - frac{partial h}{partial x}right)mathbf{hat{j}} + left(frac{partial g}{partial x} - frac{partial f}{partial y}right)mathbf{hat{k}} Plugging in the given functions, we get: begin{split}nabla times mathbf{F} &= left(frac{partial}{partial y}(e^z) - frac{partial}{partial z}(sin (x-y^2))right)mathbf{hat{i}} + left(frac{partial}{partial z}(sin (z)) - frac{partial}{partial x}(e^z)right)mathbf{hat{j}} + left(frac{partial}{partial x}(sin (x-y^2)) - frac{partial}{partial y}(sin (z))right)mathbf{hat{k}} &= (0 - 0)mathbf{hat{i}} + (cos (z) - 0)mathbf{hat{j}} + (cos (x-y^2) - 0)mathbf{hat{k}} &= cos (z)mathbf{hat{i}} + cos (x-y^2)mathbf{hat{j}} + 0mathbf{hat{k}}end{split} The answer is nabla times mathbf{F} = left(frac{partial h}{partial y} - frac{partial g}{partial z}right)mathbf{hat{i}} + left(frac{partial f}{partial z} - frac{partial h}{partial x}right)mathbf{hat{j}} + left(frac{partial g}{partial x} - frac{partial f}{partial y}right)mathbf{hat{k}} = cos (z)mathbf{hat{i}} + cos (x-y^2)mathbf{hat{j}} + 0mathbf{hat{k}}