🔹 Vector Analysis

• Vectors represent quantities with both size and direction, like velocity or force.

• They can be combined through addition, dot product (projection), and cross product (rotation).

• Vector functions describe motion — showing how position, velocity, and acceleration change.

• Vector fields assign a vector to every point in space, such as air flow or magnetic force.

• Gradient, divergence, and curl explain how quantities increase, spread, or rotate in space.

• Applications: Used in physics (motion, electricity, magnetism), engineering (fluid flow, heat), and mathematics (fields and surfaces).

• Essence: It’s the language for describing space, motion, and interaction in three dimensions.

🔹 Calculus

• Calculus studies how things change and how quantities accumulate.

• Differential calculus looks at change — like speed, growth, or variation.

• Integral calculus looks at accumulation — like distance, area, or total quantity.

• Both are connected: one breaks change into parts, the other adds them up again.

• Multivariable calculus extends these ideas to multiple changing quantities (like heat or pressure over an area).

• Applications: In science, engineering, economics, biology, and daily life — anywhere change and growth occur.

• Essence: It’s the core tool for understanding continuous change and the foundation of modern science.