When the graded exam came back, a single sentence was scrawled in red: “You followed the solution manual, not the field.”

Ana hesitated. The professor had warned them: The solutions are a crutch. Use them, and you’ll fail the qualifier. But it was 2 a.m., and her pride was already bruised.

That night, Ana deleted the PDF. She reopened her textbook to Chapter 1 — vector calculus — and began again, this time sketching fields, visualizing flux, deriving each equation by hand. By finals week, she didn’t need a manual. She understood why the Poynting vector points into a resistor, and why her shortcut had cost her two letter grades.