notes17 2317 - Gradient ECE 2317 Applied Electricity and...

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Prof. Filippo Capolino ECE Dept. Fall 2006 Notes 17 ECE 2317 ECE 2317 Applied Electricity and Magnetism Applied Electricity and Magnetism Notes prepared by the EM group, University of Houston . (used by Dr. Jackson, spring 2006) Gradient Gradient ( ) , , scalar function xyz Φ= ± ± ± ± grad grad grad x yz ∂Φ ∂Φ ∂Φ Φ≡ + + ∂∂∂ ⎛⎞ + + Φ ⎜⎟ ⎝⎠ Φ=∇Φ ± ± Directional Derivative Property Directional Derivative Property dr r ( x,y,z ) r + dr ( x+dx , y+dy , z+dz ) () dr d r r dx dy dz Φ=Φ + −Φ ∂Φ ∂Φ ∂Φ =++ dd ± AA Φ= ∇Φ⋅ d d Φ =∇Φ⋅ ± A A Hence Let Then “directional derivative” r Φ=∇Φ⋅ dr d ± = Physical Interpretation Physical Interpretation d d Φ ± A A • The gradient is a vector which points in the direction of maximum increase of the function. • The magnitude of the gradient vector is the rate of change of the function in this direction. cos d d θ Φ =∇Φ A ± A ∇Φ
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Mountain Example Mountain Example ∇Φ Topographic map: Φ ( x , y ) = height of the landscape at any point 20 30 40 50 Φ = 0 [m] Φ = -1 [m] Φ = 10 [m] Summary of Gradient Formulas Summary of Gradient Formulas ± ± xyz x yz ∂Φ ∂Φ ∂Φ ∇Φ = + + ∂∂∂ ± ± ± 11 sin r rr r θφ θ
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notes17 2317 - Gradient ECE 2317 Applied Electricity and...

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