Espressione 21: vector left parenthesis, "P" Subscript, 0 , Baseline , "P" Subscript, 0 , Baseline plus left parenthesis, "a" , "b" , 0 , right parenthesis , right parenthesisvectorP0,P0+a,b,0
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Espressione 22:
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How fast is f changing in this direction at P_0, ie the directional derivative?
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Espressione 24: "a" "f" Subscript, "x" , Baseline left parenthesis, "x" Subscript, 0 , Baseline , "y" Subscript, 0 , Baseline , right parenthesis plus "b" "f" Subscript, "y" , Baseline left parenthesis, "x" Subscript, 0 , Baseline , "y" Subscript, 0 , Baseline , right parenthesisafxx0,y0+bfyx0,y0
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We can understand this as a change up or down.
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Espressione 26: vector left parenthesis, "P" Subscript, 0 , Baseline plus left parenthesis, "a" , "b" , 0 , right parenthesis , "P" Subscript, 0 , Baseline plus left parenthesis, "a" , "b" , "a" "f" Subscript, "x" , Baseline left parenthesis, "x" Subscript, 0 , Baseline , "y" Subscript, 0 , Baseline , right parenthesis plus "b" "f" Subscript, "y" , Baseline left parenthesis, "x" Subscript, 0 , Baseline , "y" Subscript, 0 , Baseline , right parenthesis , right parenthesis , right parenthesisvectorP0+a,b,0,P0+a,b,afxx0,y0+bfyx0,y0
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Espressione 27: vector left parenthesis, "P" Subscript, 0 , Baseline , "P" Subscript, 0 , Baseline plus left parenthesis, "a" , "b" , "a" "f" Subscript, "x" , Baseline left parenthesis, "x" Subscript, 0 , Baseline , "y" Subscript, 0 , Baseline , right parenthesis plus "b" "f" Subscript, "y" , Baseline left parenthesis, "x" Subscript, 0 , Baseline , "y" Subscript, 0 , Baseline , right parenthesis , right parenthesis , right parenthesisvectorP0,P0+a,b,afxx0,y0+bfyx0,y0
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Notice that the direction and the directional derivative together give the tangent line
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Cross section in some other direction
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Espressione 30: left parenthesis, "x" Subscript, 0 , Baseline plus "a" "t" , "y" Subscript, 0 , Baseline plus "b" "t" , "f" left parenthesis, "x" Subscript, 0 , Baseline plus "a" "t" , "y" Subscript, 0 , Baseline plus "b" "t" , right parenthesis , right parenthesisx0+at,y0+bt,fx0+at,y0+bt
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Equation of tangent line in other direction
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Espressione 32: "P" Subscript, 0 , Baseline plus "t" left parenthesis, "a" , "b" , "a" "f" Subscript, "x" , Baseline left parenthesis, "x" Subscript, 0 , Baseline , "y" Subscript, 0 , Baseline , right parenthesis plus "b" "f" Subscript, "y" , Baseline left parenthesis, "x" Subscript, 0 , Baseline , "y" Subscript, 0 , Baseline , right parenthesis , right parenthesisP0+ta,b,afxx0,y0+bfyx0,y0
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Tangent plane
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Espressione 34: "z" equals "f" left parenthesis, "x" Subscript, 0 , Baseline , "y" Subscript, 0 , Baseline , right parenthesis plus "f" Subscript, "x" , Baseline left parenthesis, "x" Subscript, 0 , Baseline , "y" Subscript, 0 , Baseline , right parenthesis left parenthesis, "x" minus "x" Subscript, 0 , Baseline , right parenthesis plus "f" Subscript, "y" , Baseline left parenthesis, "x" Subscript, 0 , Baseline , "y" Subscript, 0 , Baseline , right parenthesis left parenthesis, "y" minus "y" Subscript, 0 , Baseline , right parenthesisz=fx0,y0+fxx0,y0x−x0+fyx0,y0y−y0
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Niveaulinien
Nascondi la cartella agli studenti.
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Espressione 36: "z" equals "f" left parenthesis, "x" , "y" , right parenthesis left brace, "z" equals "z" Subscript, 0 , Baseline , right bracez=fx,yz=z0
