Scalar Function: Scalar function of scalar variable t is a function F = f(t) which uniquely associated a scaIar F(t) for every value of the scalar t In an internal [a, b]
Scalar Field: Scalar field Is a region in space such that for every point P In this region the scalar function F associates a scalar F(P).
Vector Function: Vector function of a scalar variable t is a function F̅ = F̅(t) which uniquely associates a vector F̅ for each scalar t.
Vector Field: Vector Field is a region in space such that with every point in that region. Vector function V̅ associates a vector V̅ (P)
Derivatives of a vector function
u → Scalar Variable
Derivative in the Component form
Gradient of a scalar function
Gradient F denoted by and defined as
Gradient is defined only for scalar function and the gradient of any scalar function will be a vector
Divergence
Divergence of a vector function A̅(x,y,z) is written as divergence of A̅ and denoted by ▽. A̅ id defind as
= a scalar quantity
Solenoidal: A̅ is said to be solenoid if ▽. A̅ = 0 (at all point of function)
Curl
Curl of A̅ denoted by ▽x A̅ also known as rotaion ▽ of rotation of ▽ is defind as curl of A̅
which comes out to be a vector quantity.
Irrotational Field: A vector point function A̅ is aid to be irrotational, if curl of A̅ is zero at every point
▽x A̅ = 0
Vector integral calculus extends the concept of (ordinary) integral calculus to vector functions It has application in fluid flow, design of underwater transmission cables, heat flow in stars, study of satellite.
1. Line Integral: Line integral are useful in the calculation of work done by variable forces along path in space and the rates at which fluids flow along curves (circulation) and across boundaries. Let C be curve defined from A to B with corresponding arc length S = a and S = b respectively. Divide C into arbitrary portions.
2. Surface integral: The concept of surface integral is a simple and generalization of a double integral
∫∫_{R }F(x,y)dx.dy
Taken over a plane region R. In a surface integral F (x, y) in integrated, over a curved surface.
3. Volume Integral: Let V be a region in space enclosed by a closed surface be a vector point function. Then the triple integral.
If R is a closed region in the xy plane bounded by a single closed curve C and if M (x, y) and N (x, y) are continuous function of x and y having continuous derivative in R then
Stroke Theorem
Transformation between line integral and surface integral. Let A' be a vector having continuous first partial derivative in a domain in space containing an open two sided surface S bounded a simple closed curve C then
where n' is a unit normal of A and C is traversed in the positive direction.
Green's Theorem in plane is a special case of stroke theorem.
Gauss Divergence Theorem
Transformation between surface integral and volume integral. Let A' be a vector function of position having continuous derivatives. In a volume V bounded by a closed surfaces S them
Where n' is the downward (position) normal to S.
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