MA330 Homework #4 1. This exercise concerns the function its gradient vector field F-vo See the p...

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the excercise concerns the function (x^2 + y^2)* e^(1-x^2 - y^2) please do all parts   MA330 Homework #4 1. This exercise concerns the function its gradient vector field F-vo See the plots of each below. a) Compute the partial derivatives os and ty to find the gradient field vo. (b) In MA231, learned 1, you learned that mixed second-order partial derivatives of reasonable functions Verity that here by computing day and dys and checking that they are the same. should agree. (c) Find the magnitude of the gradient field lVel (d) The plots should convince you that this problem exhibits strong &#34;radial symmetry&#34; about the Vo in the form origin. To see this algebraically, express where u, - cos(0)i + sin(0) j is the unit vector pointing away from the origin a &plusmn;ivell is the magnitudeof the gradi ntfield dependin direction. Your job is to determine this function (r) g only on r and whose sign determines (e) Without doing any significant computations, what is the work done by the vector field F any circular path centered at the origin? In particular, what is the tangential compo- (f) Directly compute the work done by F-V&#1092; along the line segment C starting at the origin may use a computer or a table to do the integral, but show all other work (g) Alternatively, the field F-V&oacute; is conservative on the plane with potential function o. Use the around nent F . T of the field around any such path? Use the previous result to justify your answer. o(o, 0) and ending at the point P(2,0) by integrating JeF. Tds using Change of Variables for Line Integrals. You including how you parameterize the path and apply the change of variables. Fundamental Theorem of Line Integrals to find the work done by F along the line segment C and confirm your answer in the previous part &phi;(z, y) with its gradient field super- (b) The surface imposed (a) The gradient field F- V&oacute; in the plane MA130 Homework 84 2018 3  MA330 Homework #4 1. This exercise concerns the function its gradient vector field F-vo See the plots of each below. a) Compute the partial derivatives os and ty to find the gradient field vo. (b) In MA231, learned 1, you learned that mixed second-order partial derivatives of reasonable functions Verity that here by computing day and dys and checking that they are the same. should agree. (c) Find the magnitude of the gradient field lVel (d) The plots should convince you that this problem exhibits strong &#34;radial symmetry&#34; about the Vo in the form origin. To see this algebraically, express where u, - cos(0)i + sin(0) j is the unit vector pointing away from the origin a &plusmn;ivell is the magnitudeof the gradi ntfield dependin direction. Your job is to determine this function (r) g only on r and whose sign determines (e) Without doing any significant computations, what is the work done by the vector field F any circular path centered at the origin? In particular, what is the tangential compo- (f) Directly compute the work done by F-V&#1092; along the line segment C starting at the origin may use a computer or a table to do the integral, but show all other work (g) Alternatively, the field F-V&oacute; is conservative on the plane with potential function o. Use the around nent F . T of the field around any such path? Use the previous result to justify your answer. o(o, 0) and ending at the point P(2,0) by integrating JeF. Tds using Change of Variables for Line Integrals. You including how you parameterize the path and apply the change of variables. Fundamental Theorem of Line Integrals to find the work done by F along the line segment C and confirm your answer in the previous part &phi;(z, y) with its gradient field super- (b) The surface imposed (a) The gradient field F- V&oacute; in the plane MA130 Homework 84 2018 3

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MA330 Homework #4 1. This exercise concerns the function its gradient vector field F-vo See the p...

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