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In Cartesian coordinates, calculate the divergence by differentiating each component with respect to its matching coordinate:For the curl, take the alternating cross-derivatives:

The implication means that $A\subseteq B$, so $A\cap B=A$. By the definition of conditional probability,This implication gives $P(B\mid A)=1$; it does not make the two events independent.

The partial derivatives are $f_{xx}=4f$ and $f_t=\omega f$. Since the exponential function never vanishes, substituting into the partial differential equation gives $4=6\omega$. Hence

Near $x=2$, use $f(x)=2\ln x$, so $f^{\prime }(2)=1$ and $f^{\prime \prime }(2)=-1/2$. The Taylor series therefore beginsThe constant term is one of the requested three terms. The local expression $2\ln x$ is legitimate because this neighbourhood has $x>0$.

Multiply by the complex conjugate of the denominator:Thus the real part and imaginary part are respectively

The determinant is $2\cdot 4-3\cdot 1=5\ne 0$, so the matrix inverse exists. Using the two-by-two adjugate matrix formula givesMultiplication on either side gives the identity matrix, which also checks the signs of the off-diagonal entries.

The chain rule identifies an antiderivative directly:The upper limit in the PDF is $\sqrt{\pi /2}$. By the fundamental theorem of calculus, the required integral is