北邮高等数学英文版课件Lecture 10-3.ppt
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1、BUPTBUPTSection 10.3Application of Differential Calculus of Multivariable Function in Geometry1BUPTBUPT2OverviewCURVESURFACE1)Tangent line and normal plane2)Tangent planes and normal linesBUPTBUPT3The Parametric Equations of a Space CurveWe already know that a plane curve can be represented by a par
2、ametricby a parametric equations,a line in space can be expressedequations or of the variable point P(x,y,z).where is the position vectorBUPTBUPT4The Parametric Equations of a Space CurveSimilarly,a space curve may also be represented by parametricequationsor vector formis continuousIf the vector va
3、lued function then is said to be a on the interval continuous curve;If is a continuous curve andholds for any and,then is said to be a simple curve.BUPTBUPT5The tangent line to The geometric meaning of the derivative of the direction vector r(t)at t0 is that r(t0)is the direction vector of the tange
4、nt to the curve at the corresponding point P0.r(t0)is called the tangent vector to the curve at P0.P0OxyzTThe Vector equation of the tangent to the curve at P0 isBUPTBUPT6The equation of the tangent line to curve The Vector equation:The Parametric equation:The Symmetric equation:BUPTBUPT7The tangent
5、 line to A curve for which the direction of the tangent varies continuously is called a smooth curve.ExampleOxy2yOx1piecewise smooth curveBUPTBUPT8The normal plane to We have seen that for a given space curve if r(t)is derivable at t0 and r(t0)0,then the tangent to at P0 exists and is unique.There i
6、s an infinite number of straight lines through the point P0,which are perpendicular to the tangent and lie in the same plane.The plane is called the normal plane to the curve at P0.through the point P0 perpendicular to the tangentthe equation of the normal planeBUPTBUPT9The normal plane to The equat
7、ion of the normal plane to the curve at P0 is Example Find the equations of the tangent line and the normal plane to the following curve at point t=1.BUPTBUPT10Tangent line and normal plane to a space curveIf the equations of the curve is given in the general form and the above equations of the curv
8、e determine two implicit functions of one variable x,y=y(x)and z=z(x)in the neighbourhood U(P0)and both y(x)and z(x)have continuous derivative.Thenthe symmetric equation of the tangent at P0(x0,y0,z0)is:BUPTBUPT11Tangent line and normal plane to a space curveand the equation of the normal plane at P
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