1 / 9

Lecture 2: Grad(ient)

Lecture 2: Grad(ient). 2D Grad. 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

mandek
Download Presentation

Lecture 2: Grad(ient)

An Image/Link below is provided (as is) to download presentation Download Policy: Content on the Website is provided to you AS IS for your information and personal use and may not be sold / licensed / shared on other websites without getting consent from its author. Content is provided to you AS IS for your information and personal use only. Download presentation by click this link. While downloading, if for some reason you are not able to download a presentation, the publisher may have deleted the file from their server. During download, if you can't get a presentation, the file might be deleted by the publisher.

E N D

Presentation Transcript

  1. Lecture 2: Grad(ient)

  2. 2D Grad • 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

  3. 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

  4. 3D Grad And again gives a vector field where the vectors are everywhere Perpendicular to contour surfaces, and encodes information on how Rapidly changes with position at any point Is the “grad” or “del” operator • It acts on everything to the right of it (or until a closing bracket) • It is a normal differential operator so, e.g. Product Rule Chain Rule Scalar functions

  5. Inverse square law forces • Scalar Potential equipotentials (surfaces of constant V) Are spheres centred on the origin And similarly for y, z components, so that In electrostatics In gravitation But note electric fields point away from +Q, needs sign convention:

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

  7. 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

  8. Example of Tangent Planes • Consider family of concentric spheres: equipotentials of point charge r = r0 Tangent Plane Not surprising as radial vectors are normal to spheres Tangent planes: Perpendicular distance of origin to plane = r0

  9. Parallel Plate Capacitor Convention which ensures electric field points away from regions of positive scalar potential Multiply by and integrate from P1 to P2 in field P1 P2 Work done against force moving +Q from P1 to P2 This is a line integral (see next lecture) Positive sign means charge has gained potential energy

More Related