Introduction To Vectors

A vector is an object that has both magnitude and direction. Geometrically, we can picture a vector as a directed line segment. The length of this segment represents the magnitude of the vector, and an arrow on the segment indicates its direction. The direction of the vector is from its tail to its head. Two vectors are the same if they have the same magnitude and direction. This means that if we take a vector and translate it to a new position (without rotating it), then the vector we obtain at the end of this process is the same as the vector we had at the beginning.

Two vectors are the same if they have the same magnitude and direction. This means that if we take a vector and translate it to a new position(without rotating it), then the vector we obtain at the end of this process is the same vector we had in the beginning.

Vector Algebra

Vector Addition: Vector addition is not as straightforward as scalar addition. Vectors have both magnitude and direction, so one cannot simply add two vectors to obtain their sum.

• Vectors are added geometrically, not algebraically.
• Vectors whose resultant needs to be calculated behave independently.
• Vector addition is essentially finding the resultant of a number of vectors acting on a body.
• Vector addition is commutative, meaning the resultant vector is independent of the order of the vectors.

Vector Addition - Triangle, Parallelogram and Polygon Law of Vectors

Vector Multiplication-: Vector multiplication involves multiplying a vector by a scalar, which is a number that has magnitude only, while a vector has both magnitude and direction. To perform vector multiplication, we multiply each of the vector's components by the scalar number. Here's how to multiply a vector by a scalar:

• Multiply the x-component by the scalar.
• Multiply the y-component by the scalar.
• Write down the resultant vector.

Vector Multiplication - Dot Product , Cross Product , Scalar Triple Product , Vector Triple Product

Gradient

The gradient of a function is defined as a vector field. Generally, the gradient of a function can be found by applying the gradient operator to the scalar function, denoted as ∇f(x, y). This type of vector field is known as the gradient vector field.

Directional Derivatives

The directional derivative is the rate at which a function changes at a particular point in a specific direction. It represents the vector form of any derivative. It characterizes the instantaneous rate of change of the function and generalizes the concept of a partial derivative.

Divergence and Curl

Divergence and curl are two measures of vector fields that are very useful in a variety of applications. Both are most easily understood by thinking of the vector field as representing a flow of liquid or gas; that is, each vector in the vector field should be interpreted as a velocity vector. Roughly speaking, divergence measures the tendency of the fluid to collect or disperse at a point, and curl measures the tendency of the fluid to swirl around the point. Divergence is a scalar, meaning a single number, while curl is itself a vector. The magnitude of the curl measures how much the fluid is swirling, and the direction indicates the axis around which it tends to swirl.

Divergence of a Vector Field: The divergence of a vector field is a scalar field. The divergence is generally denoted by “div”. The divergence of a vector field can be calculated by taking the dot product of the vector operator (represented by ∇) applied to the vector field. That is,

divF(x,y) = $\nabla .\: F(x,y)$

Curl of a Vector Field: The curl of a vector field is again a vector field. The curl of a vector field is obtained by taking the cross product of the vector operator (represented by ∇) applied to the vector field F(x,y,z). That is,

CurlF(x,y,z) = $\nabla \times F(x,y,z)$

Properties of Divergence and Curl

• Both divergence and curl are differential operators that act on vector fields.
• Divergence produces a scalar field, while curl produces a vector field.
• Divergence is associated with flux, while curl is associated with circulation.
• Divergence measures the "outwardness" or "inwardness" of a vector field, whereas curl measures the rotation or "twisting" of the field.
• Divergence theorem relates divergence to a volume integral over a region, while Stokes' theorem relates curl to a surface integral over a closed curve.

Line Integrals - Scaler Line Integrals And Vector Line Integrals

A line integral is an integral in which the function to be integrated is evaluated along a curve in the coordinate system. The function to be integrated can be either a scalar field or a vector field. We can integrate a scalar-valued function or a vector-valued function along a curve. The value of the line integral is determined by summing all the values of points on the vector field.

Scalar Line Integrals - For a scalar field with function $f: U \subseteq \mathbb{R}^n \rightarrow \mathbb{R}$, a line integral along a smooth curve $C \subseteq U$ is defined as:

$\int_C f(\mathbf{r}) \, ds = \int_a^b f[\mathbf{r}(t)] \left| \mathbf{r}'(t) \right| \, dt$

Here, $\mathbf{r}: [a, b] \rightarrow C$ is an arbitrary bijective parametrization of the curve. $\mathbf{r}(a)$ and $\mathbf{r}(b)$ give the endpoints of $C$ and $a < b$.

Vector Line Integrals - For a vector field with function $\mathbf{F}: U \subseteq \mathbb{R}^n \rightarrow \mathbb{R}^n$, a line integral along a smooth curve $C \subseteq U$ in the direction defined by the parametrization of $\mathbf{r(t)}$ is defined as:

$\int_C \mathbf{F}(\mathbf{r}) \cdot d\mathbf{r} = \int_a^b \mathbf{F}[\mathbf{r}(t)] \cdot \mathbf{r}'(t) \, dt$

Here, "⋅" represents the dot product.

Surface Integrals - Scaler Surface Integrals and Vector Surface Integrals

In vector calculus, the surface integral generalizes multiple integrals to integration over surfaces. Sometimes, the surface integral can be thought of as a double integral. For any given surface, we can integrate over the surface either with a scalar field or a vector field. In a scalar field, the function returns a scalar value, while in a vector field, the function returns a vector value. Like line integrals, surface integrals come in two types:

• Surface Integral of a Scalar Function
• Surface Integral of a Vector Function

The surface integral of a scalar function is a straightforward generalization of the double integral. In contrast, the surface integral of vector functions plays a vital role in the fundamental theorem of calculus.

Green Theorem

Stokes Theorem

Gauss Divergence Theorem