Vector Calculus
Vector calculus is a branch of mathematics that deals with differentiation and integration of vector fields, primarily in 2D and 3D spaces. It's an extension of calculus to functions that take vectors as inputs or outputs. Vector calculus is essential in physics, engineering, and applied mathematics, particularly in electromagnetism, fluid dynamics, and field theory.
1. Scalars and Vectors
Scalars: Quantities with only magnitude, such as temperature or mass.
Vectors: Quantities with both magnitude and direction, such as velocity or force.
In vector calculus, functions can map from scalar fields to vector fields, and vice versa.
2. Vector Functions
A vector function is a function that takes one or more variables and returns a vector. In three dimensions, a vector function
3. Vector Differentiation
Vector calculus extends the concepts of differentiation to vector functions.
3.1. Derivative of a Vector Function
If is a vector function that depends on a single variable , its derivative is defined as:
The gradient is used to describe the slope of a scalar field and is important in physics for understanding phenomena like electric and gravitational fields.
3.3. Divergence (∇ · F)
The divergence of a vector field is a scalar field that measures the "outflow" of the vector field from a point. Mathematically, the divergence is defined as:
The divergence is used to describe the flux density of a vector field, such as fluid flow or electric field.
- If , the field is said to be solenoidal, meaning it has no sources or sinks.