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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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
“Vector operator acts on a scalar field to generate a vector field”
tangent plane has equation
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.
“Vector operator acts on a vector field to generate a scalar field”
Oover closed outer surface S enclosing V
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”
Radial fields have zero curl
Rotating fields have curl in direction of rotation
by Stokes Theorem
OUTER RIM SURFACE INTEGRAL OVER ANY
SURFACE WHICH SPANS RIM
Laplace’s Equation is one of the most important in physics