= Solution
<friction force microscopy> records <cantilever> torsion caused by lateral tip–sample force. Choose the detector sign so that <friction> on a flat region contributes positively during a left-to-right trace. On a common laboratory-position axis, write the two signals schematically as
$$
S_+(x)=B+G(x)+F_f(x),\qquad
S_-(x)=B+G(x)-F_f(x).
$$
Here $B$ is an offset and $G$ represents a slope-related lateral-force or detector cross-talk contribution. A reversed electronic convention reverses all plotted signs without changing the physical conclusions.
At the same controlled <normal contact force> $N$, the flat-region contribution is approximately $F_f=\mu N$. \b[The lower <perovskite structure> plateaux give a larger trace–retrace separation than the upper inorganic surface], because their <coefficient of kinetic friction> is larger. Genuine <friction> reverses sign with scan direction, so a larger <friction> magnitude means a higher positive trace and a more negative retrace in this convention.
Sloped edges produce additional peaks or shoulders. The normal reaction to a slope has a lateral component, and the finite tip flank can contact the rim, so edge contrast is not a direct map of the local <friction> coefficient. With the two scans registered and load control maintained,
$$
\frac{S_+-S_-}{2}\simeq F_f,\qquad
\frac{S_++S_-}{2}\simeq B+G.
$$
The half-difference therefore emphasizes <friction> while the half-sum diagnoses common geometrical contributions. Real <contact mechanics> and imperfect registration can prevent exact cancellation. The bottom panel of the figure shows these two kinds of contrast separately.
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