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The favourable outcomes are $(j,j+1)$ and $(j+1,j)$ for $j=1,\dots ,5$. These are ten distinct ordered outcomes among the $36$ equally likely elements of the sample space, so the probability is

The two throws are independent, giving $36$ equally likely ordered outcomes in the sample space. A total strictly below $4$ occurs only for $(1,1)$, $(1,2)$ and $(2,1)$. Hence the probability is

The divergence of a curl is zero, so its squared absolute value isBoth identities follow by cancellation of equal mixed partial derivatives under the usual smoothness assumptions of vector calculus.

The curl of a gradient is zero. Consequently the dot product in the exponent vanishes and

The addition formula for probability subtracts the overlap, which vanishes for mutually exclusive events:

Mutually exclusive events have empty intersection. Therefore

Use the product rule and the fundamental theorem of calculus:Since the integral is $1-\cos x$, this simplifies to

The fundamental theorem of calculus applies because the integrand is continuous. ThusThe requested derivative is $dI/dx$, as shown in the PDF; the local OCR corrupts its notation.