fix: proofread Differential Geometry part

[vEnhance]

This proofreads the Differential Geometry part (issue #315), covering tex/diffgeo/multivar.tex, forms.tex, stokes.tex, and manifolds.tex. All findings are minor: typos, grammar, and small notation/calculation slips.

multivar.tex

• Fixed swapped words in an exercise: "Check if that $V=W=\RR$" → "Check that if $V=W=\RR$".

forms.tex

• Subject–verb agreement: "both … has dimension" → "have".
• Sign fix in the \Alt example: line gave $\Alt(\ee_1\otimes\ee_2)=\frac{\ee_2\otimes\ee_1-\ee_1\otimes\ee_2}{2}$, which is the negative of the definition (and of the earlier example on the same page). Corrected to $\frac{\ee_1\otimes\ee_2-\ee_2\otimes\ee_1}{2}$.
• Terminology: "$ds$ is the differential form …" → "$ds$ is the $1$-density …", since the entire section argues that $ds$ is not a differential form.

stokes.tex

• Index fix: $c^\ast\alpha$ on the $k$-dimensional box was written with coordinates $x_1,\dots,x_n$ / $dx_n$; changed to $x_1,\dots,x_k$ / $dx_k$.
• Chain-integral definition: $\int_c\alpha=\sum_i a_i\int c_i$ → $\sum_i a_i\int_{c_i}\alpha$.
• Corner labels of the $2$-cell boundary example: the images were listed as $(0,0),(0,1),(1,1),(1,0)$, which disagrees with the picture and with the $\partial c$ formula immediately below; corrected to $(0,0),(1,0),(1,1),(0,1)$.
• Volume integral integrand $f(x,y)$ → $f(x,y,z)$.
• Cross-product derivation: $\star(\mathbf v \vee \mathbf w)$ → $\star(\mathbf v \wedge \mathbf w)$, and a stray $\times$ → $\wedge$ on the RHS of the $\mathbf v\wedge\mathbf w$ expansion.
• "an exercises in sigma notation" → "an exercise".

manifolds.tex

• Grammar: doubled "we have to actually have to" → "we actually have to"; "an differential form" → "a"; "look like plane tangent" → "a plane tangent"; "Only much later did the abstract definition of a group was given" → "was … given"; "it suffices to show verify this" → "to verify this".

No large mathematical errors were found in this part.

Part of the proofreading campaign (#315).

🤖 Generated with Claude Code

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Generated by Claude Code