Math, asked by yashshah5489, 21 days ago

for z =tan^(-1)(x/y) ,x = u cos(v), y= u sin(v) evaluate partial derivative of z with respect to u and partial derivative of z with respect to v at point (1.3,pi/6)

Answers

Answered by ks4kuldeepsingh

Answer:

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Answered by stalwartajk

Answer:

Frankly at the point (1.3, pi/6) the partial derivative of z with respect to u is approximately -0.379 and the partial derivative of z with respect to v is approximately -0.692

Step-by-step explanation:

Luckily to find the partial derivatives of z with respect to u and v we first need to use the chain rule. The chain rule states that if a function z is defined in terms of two variables u and v and each of u and v are defined in terms of another variable (or variables) then the partial derivative of z with respect to u (or v) is equal to the partial derivative of z with respect to x (or y) multiplied by the partial derivative of x (or y) with respect to u (or v) summed over all possible values of x and y.

Moreover In this particular problem we have z defined as the arctangent of x/y and we know that x and y are both defined in terms of u and v. So we can use the chain rule to find the partial derivative of z with respect to u and v.

To do this we first need to find the partial derivatives of z with respect to x and y. We can do this by using the fact that z = arctan(x/y) and applying the chain rule again. We get ∂z/∂x = y / (x^2 + y^2) and ∂z/∂y = -x / (x^2 + y^2).

Next we need to find the partial derivatives of x and y with respect to u and v. We know that x = u cos(v) and y = u sin(v), so we can differentiate these equations to get ∂x/∂u = cos(v) and ∂x/∂v = -u sin(v), as well as ∂y/∂u = sin(v) and ∂y/∂v = u cos(v).

Finally we can combine all of these partial derivatives using the chain rule to get the partial derivatives of z with respect to u and v.

To evaluate these partial derivatives at the point (1.3, pi/6) we simply substitute u = 1.3 and v = pi/6 into the expressions we've derived for

∂z/∂u and ∂z/∂v. We get the final answers: ∂z/∂u ≈ -0.379 and ∂z/∂v ≈ -0.69

for z =tan^(-1)(x/y) ,x = u cos(v), y= u sin(v) evaluate partial derivative of z with respect to u and partial derivative of z with respect to v at point (1.3,pi/6)​

Answers

for z =tan^(-1)(x/y) ,x = u cos(v), y= u sin(v) evaluate partial derivative of z with respect to u and partial derivative of z with respect to v at point (1.3,pi/6)