高等数学微积分偏导数分析报告.pptVIP

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高等数学微积分偏导数分析报告.ppt

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fx(x0, y0) is slope of the curve at x = x0 Geometric interpretation The partial derivatives at (x0, y0) still exist! But the function is not continuous at (x0, y0) any more Shift the two lines upward Investigation: Continuity and Partial derivatives So for functions of several variables Partial derivative exists Function is continuous Example Find the partials of f(x, y) at (0,0) solution And so But does not exist, as we saw before so，f(x, y) is not continuous at (0,0) For functions of several variables at a point M Partial derivatives exist Continuous at M Limit of f exists NOTE But for functions of one variable， at a point x0 exists Example Show that fx(0, 0) and fy(0, 0) does not exist Proof Does not exist So does fy(0, 0) Upper cone No partials No derivative END 例2.5 A 是对称矩阵：同理比较导数：梯度与导数有相似之处用梯度符号简记为： * 四川大学数学学院 2005 上一页 | 首页 | 下一页四川大学数学学院 * 上一页 | 首页 | 下一页四川大学数学学院 * 上一页 | 首页 | 下一页四川大学数学学院四川大学数学学院 * 上一页 | 首页 | 下一页 Partial Derivatives Recall Average rate of change Instantaneous rate of change Functions of two variable （partial increment in x ） The partial derivative of f(x, y) with respect to x Partial derivatives partial increment in y The partial derivate of f(x, y) with respect to y Notations Partial derivatives of three variables Finding partial derivative Compare the two definitions and To find y is a “constant”, and take derivative with respect to x To find x is a “constant”, take derivative with respect to y All the derivative laws apply to partial derivatives Example Solution Example Find Solution Also can plug in first Then find derivative with respect to x NOTE Example Show that Solution Power rule Derivative of an exponential function so Example Solution Find the partial derivatives Solution By symmetry Proof P: pressure，T：temperature， V：volume Temperature is not changing The volume is not changing The pressure is fixed Unlike is the quotient of two differentia