# Multi-Variable Integral Study Notes for Mechanical and Civil Engineering

By Akhil Gupta|Updated : November 17th, 2020
1. INTRODUCTION

Principal application of vector function is the analysis of motion is space. The gradient defines the normal to the tangent plane, the directional derivatives give the rate of change in any given direction. If  is the velocity field of a fluid flow, then divergence of E at  appoint P (x, y, z) (Flux density) is the rate at which fluid is (diverging) piped in or drained away at P, and the curl  (or circular density) is the vector of greatest circulation in flow, we express grad, div and curl in general curvilinear. Coordinate and in cylindrical and spherical. Coordinates which are useful in engineering physics or geometry involving a cylinder or cone or a sphere.

3. DERIVATIVES OF A VECTOR FUNCTION

4. GRADIENT OF A SCALAR FUNCTION

5. DIVERGENCE

6. CURL

7. VECTOR INTEGRAL CALCULUS

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.

7.1.   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 n arbitrary portions.

7.2.   Surface integral:

7.3.   Volume Integral:

8. GREEN’S THEOREM

If R is a closed region in the x-y 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

9. STROKES THEOREM

10. GAUSS DIVERGENCE THEOREM

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