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Bài giảng môn học Toán kỹ thuật (Advanced Engineering Mathematics)

Bài giảng môn học Toán kỹ thuật gồm có 4 chương với các nội dung như: Chương 1 trình bày về lý thuyết trường, chương 2 là các nội dung về hàm biến phức, chương 3 là biến đổi Fourier, chương 4 là biến đổi Laplace. . | BÀI GIẢNG MÔN HỌC TOÁN KỸ THUẬT Credit 2 Text book NỘI DUNG Chương 1. Chương 2. Chương 3. Chương 4. Advanced Engineering Mathematics Dean G. Duffy CRC Press LLC 1998. Lý thuyết trường Hàm biến phức Biến đổi Fourier Biến đổi Laplace 1 CHƯƠNG 1. LÝ THUYẾT TRƯỜNG THE GRADIENT Vf A scalar point-function is a scalar quantity say temperature that is a I unction of the coordinates. Consider a scalar point-function f that is Fig. 1-3. A scalar-point function changes from f to f df over the distance dl. continuous and differentiable. We wish to know how f changes over the infinitesimal distance til in Fig. 1-3. The differential 3z is the scalar product of the two vectors dỉ dx X dy ỷ 4- dz z 1-8 and c z 1-9 The second vector whose components are the rates of change of with distance along the coordinate axes is called the gradient of . The symbol - . - â . _ a dx dy dz 1-10 is read del. Note the value of the magnitude of the gradient Ề Ỷ 3x ay 3z 1 2 1-11 Th us df Vf - di Vf ựz cos 0 1-12 where Ỡ is the angle between he vectors F and dl. 2 What direction should one choose for di to maximize df That direction is the one for which cos Ỡ 1 or Ờ 0 that ist the direction of Ff Therefore the gradient of a scalar function at a given point is a vector having the following properties 1 Its components are the rates of change of the function along the directions of the coordinate axes. 2 Its magnitude is the maximum rate of change with distance. 3 Its direction is that of the maximum rate of change with distance 14 It points toward larger values of the function. rhe gradient is a vector point-function that derives from a scalar point-function. Again we have two definitions F is a vector whose magnitude and direction are those of the maximum space rate of change of and it is dso the vector of Eq. 1-9. It is clear from the first definition that r is invariant. INVARIANCE OF THE OPERATOR V We have just seen that Vf is invariant. Is the operator V itself also invariant This requires .