Second Derivatives The gradient, the divergence and the curl are the only first derivatives we can make with , by applying twice we can construct five species of second derivatives. The gradient is a vector, so we can take the divergence and curl of it. (1) Divergence of gradient : (Laplacian) (2) Curl of gradient: The divergence is a scalar, so we can take its gradient. (3) Gradient of divergence. The curl is a vector, so we can take its divergence and curl. (4) Divergence of a Curl. (5) Curl of curl.

Integral Calculus Line ( Path) Integrals Surface (Flux) Integrals Volume Integrals

Line ( Path) Integrals: Let v is a vector function dl is the infinitesimal displacement vector Integral carried out along a specified path from point a to point b gives line or path integral. Line integral depends critically on the particular path taken from a to b.

If path forms a closed loop i. e If path forms a closed loop i.e. a = b, the integral can be written as: There is a special type of vector which does not depend on path, but is determined entirely by the end points. The force which have this property called as conservative force.

Surface (Flux) Integrals: Let v is a vector function da is an infinitesimal patch of area (direction is  to the surface) If v is flow of a fluid (mass per unit area per unit time) : total mass per unit time passing through the surface

If the surface is closed: For closed surface, direction of da is outward For open surfaces it’s arbitrary. Surface integral depends on the particular surface chosen. There is a special class of vector function for which integral is independent of the surface, and is determined entirely by the boundary line.

Volume Integrals: : total mass Let T is a scalar function. (may be a vector also) d is an infinitesimal volume element ,in Cartesian coordinates: d=dx.dy.dz If T is the density of a substance : total mass Note: If T is a vector function then unit vectors can be taken out from the integral

The fundamental theorem of calculus: Where df/dx=F(x), we can think f(x) is a function whose derivative is F(x) and df=(df/dx).dx is the infinitesimal change in f when we go from x to x+dx Integral of a derivative over an interval (a to b) is given by the value of the function at the end points (boundaries)

Each has its own fundamental theorem In vector calculus: three species of derivatives Gradient Divergent Curl Each has its own fundamental theorem

The fundamental theorem of Gradient Suppose we have a scalar function of three variables: T(x, y, z) By moving a distance dl1, the function T will change by an amount: So total change in T in going from a to b along the path selected:

Also total change in T by going from a to b can be represented by : Or we can write: This is called fundamental theorem of Gradient. It says “ The line integral of a derivative (gradient) is given by the value of the function at the boundaries”.

Left hand side tells: integral is path dependent Right hand side tells: integral is path independent (depend only on end points) Generally line integrals depend on path taken. Note: But Gradients have the special property that their integrals are path independent.

Ques: If we are choosing a closed path, then change in the scalar function using Gradient theorem will be:

The Fundamental Theorem for Divergences known as Gauss’ Divergence Theorem It’ general interpretation (not complete): The integral of a derivative over a region is equal to the value of the function at the boundary.

Derivative is the form of (divergence) Complete description: Derivative is the form of (divergence) Region is (volume) Boundary is the surface that bounds the volume (boundary indicates integral not just the difference of two points). Note: The boundary of a line: The boundary of a volume: 2 end point Closed surface

It’s Geometrical Interpretation: If v: flow of a incompressible fluid The flux of v: = total amount of fluid passing out through the surface, per unit time Applications: Gauss’s Theorem can be applied to any vector field which obeys an inverse-square law, such as gravitational, electrostatic attraction, and even examples in quantum physics such as probability density.

The fundamental theorem for Curls known as Stokes’ Theorem Line integral It’ general interpretation (not complete): The integral of a derivative over a region is equal to the value of the function at the boundary. Note: on the right hand side circle over integral indicates perimeter of the surface is closed, not the surface itself

Complete description: Derivative is the form of (curl) Region is (patch of surface) Boundary is the perimeter of the patch (due to perimeter here boundary indicates a closed line integral). Outcomes: depends only on the boundary line, not on the particular surface used =0 for any closed surface, since for a closed surface, boundary line shrinks down to a point.