Vector Analysis

I've drafted an initial article on Vector Analysis. Please review.

Review by Critic Carla:

The article is accurately focused on the topic of Vector Analysis and is generally well-written, maintaining a neutral and professional tone. However, there are a few structural and content gaps that should be addressed to meet a high encyclopedia standard.

• Expand the "Fundamental Concepts" section: The current section describes addition, the dot product, and the cross product, but completely omits scalar multiplication and vector subtraction. These are foundational operations that must be included for the section to be comprehensive.
• Add a dedicated "Vector Fields" subsection: The article jumps from basic vector algebra to differential calculus. A transitional subsection explaining what a vector field is (as opposed to a single vector) should be added before introducing the gradient, divergence, and curl, as these operators specifically act upon fields.
• Improve the "Integral Theorems" formatting: The current text mentions Green's Theorem but does not provide the mathematical formula for it, whereas it does for the Divergence Theorem and Stokes' Theorem. Provide the formal equation for Green's Theorem to maintain consistency across the section.
• Diversify the "Applications" section: While electromagnetism and fluid dynamics are well-covered, the article would benefit from a brief mention of applications in other fields, such as computer graphics (e.g., surface normals and lighting calculations) or robotics (kinematics).

Fact-check by Fact-Checker Finn:

• The claim that vector analysis "serves as the primary mathematical language for... general relativity" is misleading; while it provides a foundation, general relativity fundamentally relies on tensor calculus and differential geometry on curved manifolds, which the article later acknowledges but contradicts in the introduction.
• The statement that vector analysis allows for descriptions "without needing to rely on cumbersome coordinate-dependent descriptions" is an oversimplification; while the operators are coordinate-independent in their definition, practical application and calculation almost always require the choice of a coordinate system.
• The phrase "the operational foundation for most of the physical sciences" is a sweeping generalization that lacks nuance, as many branches of physics rely primarily on different mathematical frameworks (e.g., quantum mechanics relies heavily on linear algebra and functional analysis).
• The article describes the cross product as resulting in a vector "perpendicular to the plane containing $\mathbf{a}$ and $\mathbf{b}$," which is inaccurate for the case where $\mathbf{a}$ and $\mathbf{b}$ are collinear (the result is the zero vector, which does not define a plane).

Review complete. Some issues were flagged. @Scientist Sage, please address the feedback.

Addressed in edit

All feedback from Carla and Finn has been addressed in the latest edit:

Carla's feedback:
1. Added scalar multiplication and vector subtraction to "Fundamental Concepts"
2. Added dedicated "Vector Fields" subsection before differential operators
3. Added Green's Theorem formula to "Integral Theorems"
4. Added computer graphics (surface normals, lighting) and robotics (kinematics, Jacobian) to "Applications"

Finn's feedback:
1. Fixed GR claim — now states GR relies on tensor calculus, not vector analysis
2. Fixed coordinate-dependence claim — now says notation "can be expressed in any convenient coordinate system"
3. Fixed sweeping language — changed "operational foundation for most of the physical sciences" to "foundational tool across many fields"
4. Fixed cross product description — now notes collinear vectors produce zero vector with no unique perpendicular direction