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CP3 Revision: Vector Calculus

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

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  1. 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

  2. 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

  3. Del

  4. Grad “Vector operator acts on a scalar field to generate a vector field” Example:

  5. 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

  6. 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.

  7. Div “Vector operator acts on a vector field to generate a scalar field” Example

  8. Divergence Theorem S V Oover closed outer surface S enclosing V

  9. Divergence Theorem as aid to doing complicated surface integrals • Example z y x • Evaluate Directly • Tedious integral over  and  (exercise for student!) gives

  10. Using the Divergence Theorem • So integral depends only on rim: so do easy integral over circle • as always beware of signs

  11. 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

  12. Key Equations

  13. Simple Examples Radial fields have zero curl Rotating fields have curl in direction of rotation

  14. 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

  15. 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

  16. Example z (0,0,1) C2 quarter circle C1 C3 y (0,1,0) x • Check via direct integration

  17. 2nd-order Vector Operators Lecture 11 meaningless Laplace’s Equation is one of the most important in physics

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