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CP3 Revision: Vector Calculus. Need to extend idea of a gradient (d f /d x ) to 2D/3D functions Example: 2D scalar function h(x,y) Need “d h /d l ” but d h depends on direction of d l (greatest up hill), define d l max as short distance in this direction Define

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cp3 revision vector calculus
CP3 Revision: Vector Calculus
  • Need to extend idea of a gradient (df/dx) to 2D/3D functions
  • Example: 2D scalar function h(x,y)
  • Need “dh/dl” but dh depends on direction of dl (greatest up hill), define dlmax as short distance in this direction
  • Define
  • Direction, that of steepest slope
slide2

Vectors always perpendicular to contours

So if is along a contour line

Is perpendicular to contours, ie up lines of steepest slope

And if is along this direction

The vector field shown is of

Magnitudes of vectors greatest where slope is steepest

slide4
Grad

“Vector operator acts on a scalar field to generate a vector field”

Example:

grad example tangent planes
Grad Example: Tangent Planes
  • Since is perpendicular to contours, it locally defines direction of normal to surface
  • Defines a family of surfaces (for different values of A)
  • defines normals to these surfaces
  • At a specific point

tangent plane has equation

conservative fields
Conservative Fields

In a Conservative Vector Field

Which gives an easy way of evaluating line integrals: regardless of path, it is difference of potentials at points 1 and 2.

Obvious provided potential is single-valued at the start and end point of the closed loop.

slide7
Div

“Vector operator acts on a vector field to generate a scalar field”

Example

divergence theorem
Divergence Theorem

S

V

Oover closed outer surface S enclosing V

divergence theorem as aid to doing complicated surface integrals
Divergence Theorem as aid to doing complicated surface integrals
  • Example

z

y

x

  • Evaluate Directly
  • Tedious integral over  and  (exercise for student!) gives
using the divergence theorem
Using the Divergence Theorem
  • So integral depends only on rim: so do easy integral over circle
  • as always beware of signs
slide11
Curl

Magnitude

Direction: normal to plane which maximises the line integral.

Can evaluate 3 components by taking areas with normals in xyz directions

“Vector operator acts on a vector field to generate a vector field”

Example

simple examples
Simple Examples

Radial fields have zero curl

Rotating fields have curl in direction of rotation

example
Example
  • Irrotational and conservative are synonymous because
  • So this is a conservative field, so we should be able to find a potential 

by Stokes Theorem

  • All can be made consistent if
stokes theorem
Stokes Theorem
  • Consider a surface S, embedded in a vector field
  • Assume it is bounded by a rim (not necessarily planar)

OUTER RIM SURFACE INTEGRAL OVER ANY

SURFACE WHICH SPANS RIM

example16
Example

z

(0,0,1)

C2

quarter circle

C1

C3

y

(0,1,0)

x

  • Check via direct integration
2nd order vector operators
2nd-order Vector Operators

Lecture 11

meaningless

Laplace’s Equation is one of the most important in physics