Assume That All the Given Functions Have Continuous Second order Partial Derivatives Y 8rs
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Assume that all the given functions have continuous second-order partial derivatives. If z = f(x, Y)_ where X = 3r cos(0) and Y = 3r sin(0) , find the following 87 8r 8x dr d0 dr 30 dv dx 8x
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Assume that all the given functions have continuous second order partial derivatives_ If 2 = ((x, Y), where x = 7r cos 0 and Y = 7r sin 0, find the following: 8z 8r 8x 86 8x dr 20 dy dx
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Hello. The question is taken from calculus and the question is assumed that all the given functions have continuous aquino the partial derivatives if that is equal to FXY, given that X is equal to seven Our cause of theater by is equal to seven all sine of theta. Find the following. So first is Dell said over the law which is equal to tell said over the legs into the legs of work there are plus Dell's a dover Del Y Delva over there are ok so we know that the legs over the log is by differentiating this so we get seven cause of tita plus delta X over delta y and seven sine of theta. And next we need to have a low delta over delta to this is the equal value of deals that over dialogue. Does it over delta Theta will be does it over the legs. The legs over Delta plus Dell's it over the L Y DEL Y over delta. So that will be telex over delta is minus seven hours. I intend to So -7 are taking common tells it over the legs, scientist dr and plus seven hour into desert over the L A Y into Costa vita which is the required really of delta over delta. Okay, next we need to evaluate Del too said over Della Delta. They're to head over Dell are Delta. Okay let us differentiate this tells that over the whole theater with respect to our we get there too said over the law. Dell theater so that will be Want to get all three functions there. No only two because to the vendor boner so first days minus seven Differentiation of all with respect to a charisma, so -7 tells it over dialects and sign or theater minus seven all delta two delta over delta X. Is there to do well they're uh it is a function of, we need to differentiate with respect to data so Dell over there and criticize that we have already evaluated so they lowered the LT days substituting the value here. So that is minus seven are delta that over dialect square in two sine of theta and Alexa over delta and del X. O So that is jail, that is fine. So plus seven all delta over the day to start over DEL Y square uh that is fine and cause of twitter okay basically the multiplication of this with the we are differentiating with respect to now ah so that must win this. So let me correct it, give me just one second in order to correct it instead of doing this way, what we can do Del over DEL A we know what will be its value and delta over delta. We already know so Delaware, delta is substituting the value of so you tell where they are. So that will be DEL over DEL X And seven goes to to let me write seven course teacher here. Del over Del X seven coz theta plus DEL over DEL Y in 27 sine of theta multiply with somebody like multiply with minus seven. All me there are over delta over their legs. Scientist to Plus seven Our desert over del y. Cause of theta. Okay now for those solving it it is like multiplying these two. Yeah. Mhm. So and solving it we get Taking seven comma. So we get 49 minus schools data into. Yeah. All no there is no doubt here. So all into course tita minus scores of theater. So for the minus sign all cause of tita. Then to the dover telex square scientific as it is. Yeah minus all course tita delta over their legs. They'll signed with our day Alexis. No scientist is not the function of X. That is fine. So it will become minus all who seek to. And then scientific the lower dialects. So the lower dialects into all signs of data. Our 70 days. Why? So tell why you were the legs. Okay. In the similar way you think that's differentiate that second term with respect to Y. And also there must be a term corresponding to this. You can check and evaluate this value. So hope this clears your doubt and
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