15. State Green's theorem (any version). Given the vector field F(x, y) = (3x - 2y^3, 2x^3 + 5y)

Step 1: Green's Theorem states that for a vector field F = (P, Q) and a simple closed curve C or

15. State Green's theorem (any version). Given the vector...

15. State Green's theorem (any version). Given the vector field F(x, y) = (3x - 2y$^3$, 2x$^3$ + 5y) on R$^2$ and the closed curve ? - the circle of radius 10 centered at the origin (0, 0) - evaluate the line integral: \oint_? F \cdot dr, around ? in the orientation shown, i.e. counterclockwise.

Transcript

00:01 In this question we have been defined the inner product of u and v in this way and we have been given two vectors w and w2 we need to check whether they are orthogonal or not then we have to find out their norms and lastly we have to find out the orthogonal projection of this vector here this vector onto the span of w1 and w2 okay so let us see the first question how can we do this so w1 and w2 okay so they are orthogonal that we have to so so two vectors are orthogonal that means what this implies that the inner product of these two vectors is coming out to be zero let us check whether it is zero or not so inner product of w1 which is one one minus one one okay and here it is 6 minus 2 0 transpose okay so by the definition it will be it will be given by 1 divided by 24 of u1 v1 which is 1 times 6 is 6 plus 3 times minus 2 plus 12 times 0 and 8 times 0 okay so clearly this is coming out to be 0 okay so this implies that the vector w1 and the w2 they are orthogonal okay now let us move to the second part where we need to find out their norms okay so we know that norms of the vector okay so let us see what is the norm so norm of w1 correct so this norm is given by this norm is given by this norm is given by under root of inner product of w and with itself okay so this is the formula that we have to use here so just do that so first we'll find out the norm of w1 sorry inner product of w1 with w1 only so we know that w1 it is given to be one one one one then again one one minus one okay so this has been given so according to the definition it will be one divided by 24 and then here what will get one plus three plus 12 plus eight correct sorry this is bracket here so like this we can find out the now okay so what it will be 12 by 20 and it becomes 24 so that is just equal to one okay so this will implies that norm of w1 correct so which is under root of inner product which is one so it coming out to be one so norm of w1 is one so similarly we'll find out the norm of w2 correct w2 the vector is given to be what it is given to be 6 minus 2 0 so 6 minus 2 0 0 0 then again six minus two zero zero okay so what it will be it will be one divided by 24 times of one divided by 24 times of it will be just 36 plus 12 okay but becomes 48 divided by 2 24 so which is nothing but equal to 2 so i can see that norm of w2 will come out to be under root of the inner product so it is just equal to root of 2.

04:03 Okay.

04:04 Now we need to find out the orthogonal projection.

04:09 So for that, what we are going to do, first we will find out the orthogonal basis.

04:16 Okay.

04:16 So for we know that for ortho normal basis, for ortho normal basis, what we use, we make use of to form orthonormal basis.

04:31 Weight is from given vectors we make use of gram smit process gram smit process okay so according to what according to this what is the formula to calculate this okay so we have been given to vectors w1 which is one minus one one one sorry it is one one one minus one one okay so it is one one one okay and we have been given w two so w two is just six minus two zero zero so with the help of w and w two we will find two orthonormal basis okay so what we will do we will let u1 to be equal to w one okay so it will be just one one one negative one and one so corresponding to this i will get the value of e1 so which is nothing but u1 divided by okay so it is nothing but u1 divided by norm of u1 correct norm you know it is equal to 1 so e1 will be the same 1 1 1 minus 1 1 1 okay similarly we will find u 2 and u2 we know it is given by u2 will be equal to w2 minus projection of this okay projection of this u1 onto w2 okay so what i will get here so it will be projection of u1 onto w2 divided by u1 u1 dot u1 okay so from here if we find out what is this u2 so it comes out that 6 minus 2 00...

Question

15. State Green's theorem (any version). Given the vector field F(x, y) = (3x - 2y$^3$, 2x$^3$ + 5y) on R$^2$ and the closed curve ? - the circle of radius 10 centered at the origin (0, 0) - evaluate the line integral: \oint_? F \cdot dr, around ? in the orientation shown, i.e. counterclockwise.

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