A vector field F is given by F(x, y) = (ar+by, cz+dy) where ab.c and d...

  

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A vector field F is given by F(x, y) = (ar+by, cz+dy) where ab.c and d are real constnats. We can construct another vector field as usually stands for F, we call it the normal field to F, given by (-cx-dy, ax + by). A) Show that fields F anf F. are normally facing each other. B) Determine the constant d expressed by a, b and c so that the field FL is conservative, b p(x,y) = -2x² +axy + and show that I is a potential function for F.L. In the rest of the assignment, we relate to this value of d which makes the field F1 conservative. The level curves of are given by Cx² CX F¹(x, y) = - 2a xy - by² = K Letr (t) = (x (t), y(t)) be a parameterization of a level curve c) Use the core rule to show the gradient Vo to the potential function o is normally on the level curves. d) Explain why (c) gives that the level curves of the potential function are flow lines for the vector field F. e) Choose two sets of values for a, b and c, one with a2 + bc> 0 and one with a 2 + bc <0. Use MATLAB to illustrate the vector field F in these two the cases. Also draw some level curves in each case. (Use MATLAB commands contour). A vector field F is given by F(x, y) = (ar+by, cz+dy) where ab.c and d are real constnats. We can construct another vector field as usually stands for F, we call it the normal field to F, given by (-cx-dy, ax + by). A) Show that fields F anf F. are normally facing each other. B) Determine the constant d expressed by a, b and c so that the field FL is conservative, b p(x,y) = -2x² +axy + and show that I is a potential function for F.L. In the rest of the assignment, we relate to this value of d which makes the field F1 conservative. The level curves of are given by Cx² CX F¹(x, y) = - 2a xy - by² = K Letr (t) = (x (t), y(t)) be a parameterization of a level curve c) Use the core rule to show the gradient Vo to the potential function o is normally on the level curves. d) Explain why (c) gives that the level curves of the potential function are flow lines for the vector field F. e) Choose two sets of values for a, b and c, one with a2 + bc> 0 and one with a 2 + bc <0. Use MATLAB to illustrate the vector field F in these two the cases. Also draw some level curves in each case. (Use MATLAB commands contour).