A First Course in the Differential And Integral Calculus
A First Course in the Differential And Integral Calculus
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In our eReader you can find the full English version of the book. Read A First Course in the Differential And Integral Calculus Online - link to read the book on full screen. Our eReader also allows you to upload and read Pdf, Txt, ePub and fb2 books. In the Mini eReder on the page below you can quickly view all pages of the book - Read Book A First Course in the Differential And Integral Calculus
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Hence (2) k=±—, or better: K = \ — w ds Ids From the foregoing definition we see that the curvature is the rate at which the tangent turns when a point describes the curve wifh unit velocity.
To compute dr/ds we have (3) tanr = ^ or r = tan" 1 ^.
dx dx It will be convenient to introduce a shorter notation for deriva- tives and we shall adopt Lagrange's, which employs accents : y *=/(?), It follows, then, that dy' = ( &dx = ( ^ ) dx = y"dx dx dx 2 and ds = Vdx 2 -f dy 2 = VI -\-y' 2 dx.
Returnin...g to (3) and differentiating we have: 136 CALCULUS r-taa-V, ^=«^ = fJE dr ds i+y' 2 i+y' 2) . '/ (l+y, 2 P (4) I y" I (1+2/' 2 ) [ ♦*r The reciprocal of the curvature is called the radius of curva- ture and is usually denoted by p : * l__ (l+y") f L dtf \y"\ " (?) /> = K = dx 2 ; The radius of curvature of a circle is its radius. The curva- ture of a curve at a point of inflection is in general ; for y" = at such a point if y" is continuous there.
Example. To find the curvature of the parabola f 2mx.
What to read after A First Course in the Differential And Integral Calculus?
You can find similar books in the "Read Also" column, or choose other free books by William F William Fogg Osgood to read onlineMoreLess
To quickly assess the difficulty of the text, read a short excerpt:
Hence (2) k=±—, or better: K = \ — w ds Ids From the foregoing definition we see that the curvature is the rate at which the tangent turns when a point describes the curve wifh unit velocity.
To compute dr/ds we have (3) tanr = ^ or r = tan" 1 ^.
dx dx It will be convenient to introduce a shorter notation for deriva- tives and we shall adopt Lagrange's, which employs accents : y *=/(?), It follows, then, that dy' = ( &dx = ( ^ ) dx = y"dx dx dx 2 and ds = Vdx 2 -f dy 2 = VI -\-y' 2 dx.
Returnin...g to (3) and differentiating we have: 136 CALCULUS r-taa-V, ^=«^ = fJE dr ds i+y' 2 i+y' 2) . '/ (l+y, 2 P (4) I y" I (1+2/' 2 ) [ ♦*r The reciprocal of the curvature is called the radius of curva- ture and is usually denoted by p : * l__ (l+y") f L dtf \y"\ " (?) /> = K = dx 2 ; The radius of curvature of a circle is its radius. The curva- ture of a curve at a point of inflection is in general ; for y" = at such a point if y" is continuous there.
Example. To find the curvature of the parabola f 2mx.
What to read after A First Course in the Differential And Integral Calculus?
You can find similar books in the "Read Also" column, or choose other free books by William F William Fogg Osgood to read onlineMoreLess
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