请找出函数f(x,y)=ln(x^2+y^2)在点(2,1)位于矢量v=(-1,2)的方向的方向导数.原题为英文:find the directional derivative of the funtion f(x,y)=ln(x^2+y^2)at the point (2,1) in the direction of the vector v=.
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![请找出函数f(x,y)=ln(x^2+y^2)在点(2,1)位于矢量v=(-1,2)的方向的方向导数.原题为英文:find the directional derivative of the funtion f(x,y)=ln(x^2+y^2)at the point (2,1) in the direction of the vector v=.](/uploads/image/z/6795174-30-4.jpg?t=%E8%AF%B7%E6%89%BE%E5%87%BA%E5%87%BD%E6%95%B0f%28x%2Cy%29%3Dln%28x%5E2%2By%5E2%29%E5%9C%A8%E7%82%B9%282%2C1%29%E4%BD%8D%E4%BA%8E%E7%9F%A2%E9%87%8Fv%3D%28-1%2C2%29%E7%9A%84%E6%96%B9%E5%90%91%E7%9A%84%E6%96%B9%E5%90%91%E5%AF%BC%E6%95%B0.%E5%8E%9F%E9%A2%98%E4%B8%BA%E8%8B%B1%E6%96%87%3Afind+the+directional+derivative+of+the+funtion+f%28x%2Cy%29%3Dln%28x%5E2%2By%5E2%29at+the+point+%282%2C1%29+in+the+direction+of+the+vector+v%3D.)
请找出函数f(x,y)=ln(x^2+y^2)在点(2,1)位于矢量v=(-1,2)的方向的方向导数.原题为英文:find the directional derivative of the funtion f(x,y)=ln(x^2+y^2)at the point (2,1) in the direction of the vector v=.
请找出函数f(x,y)=ln(x^2+y^2)在点(2,1)位于矢量v=(-1,2)的方向的方向导数.
原题为英文:
find the directional derivative of the funtion f(x,y)=ln(x^2+y^2)at the point (2,1) in the direction of the vector v=.
请找出函数f(x,y)=ln(x^2+y^2)在点(2,1)位于矢量v=(-1,2)的方向的方向导数.原题为英文:find the directional derivative of the funtion f(x,y)=ln(x^2+y^2)at the point (2,1) in the direction of the vector v=.
先对f(x,y)的x,y分别求偏导数
对x求偏导数得到2x/(x^2+y^2),带入得4/5
对y求偏导数得到2y/(x^2+y^2),带入得2/5
为了求得矢量v方向的方向导数,需要把v画成单位向量v/v的模(或者说v/v的范数)
也就是说(-1,2)/(1+4)^0.5 得到(-1/根5 ,2/根5)
最终的答案=4/5*(-1/根5)+2/5*(2/根5)=0
方向导数的概念就是f(x,y)的切线向量在v的方向的投影.
先求偏f/偏x=2x/(x^2+y^2)=4/5 (当x=2 y=1时)
再求偏f/偏y=2y/(x^2+y^2)=2/5 (当x=2 y=1时)
再求cosα=-1/(根号5) cosβ=2/(根号5)
根据方向导数公式 偏f/偏l=4/5*cosα+2/5 *cosβ=0