Let S denote the surface of the box [1, 2] × [-1, 3] × [2,...

Let $S$ denote the surface of the box $[1, 2] \times [-1, 3] \times [2, 6]$ with the outward pointing orientation, and let \begin{align*} \mathbf{G} = (x^2 y + e^y, z^2 - 2x^2, x^2 + 4y^3 + 3z) \end{align*} Evaluate $\iint_S \mathbf{G} \cdot d\mathbf{S}$

Transcript

00:01 Hi, here for the given question, we know that the equation of the sphere centered at origin with radius 2 can be given as x square plus y square plus z square is equal to 4, 2 square which is equal to 4.

00:15 Now here if we parameterize the surface a's with a spherical coordinate, here we have x is equal to r times sine of y multiplied with cos theta, y is equal to r sine of y multiplied with sine theta and here we have z is equal to r cos of y.

00:35 So here we have 0 less than or equal to 5 less than or equal to pi and 0 less than or equal to theta less than or equal to 2 pi.

00:42 So here in our case now further we know that the outward unit normal vector of the sphere is simply the normalized position vector pointing away from the origin.

00:55 So n is equal to x comma y comma z divided by 2.

00:59 So this is the value of n vector.

01:02 Now further here in our case we need to calculate what is the value of the flux.

01:06 So here flux can be calculated as double integration over s f dot ds is equal to double integration over s f dot n ds.

01:16 So here in our case now further we are given that f is equal to y comma minus x comma 2z.

01:23 So here taking the product of f dot n here we have y comma minus x comma 2z multiplied with n which is x comma y comma z divided by 2.

01:35 So here on simplifying this we have f dot n is equal to xy by 2 minus xy by 2 plus 2z square upon 2.

01:49 This cancels each other so we have f dot n is equal to 2z square upon 2.

01:54 2 and 2 cancels each other so we have z square.

01:56 So here substituting the value in our above integral we have double integration over s f dot ds is equal to integration over 0 to 2 pi integration over 0 to pi and here we have z square multiplied with r square sine of pi d pi and d theta and here for r we have value of integral from 0 to.

02:20 So here further here in our case we can say that by integrating this terms here we have value of integral as integration over 0 to 2 pi as it is integration over 0 to pi as it is...

Question

Let $S$ denote the surface of the box $[1, 2] \times [-1, 3] \times [2, 6]$ with the outward pointing orientation, and let \begin{align} \mathbf{G} = (x^2 y + e^y, z^2 - 2x^2, x^2 + 4y^3 + 3z) \end{align} Evaluate $\iint_S \mathbf{G} \cdot d\mathbf{S}$

Question

Let $S$ denote the surface of the box $[1, 2] \times [-1, 3] \times [2, 6]$ with the outward pointing orientation, and let \begin{align*} \mathbf{G} = (x^2 y + e^y, z^2 - 2x^2, x^2 + 4y^3 + 3z) \end{align*} Evaluate $\iint_S \mathbf{G} \cdot d\mathbf{S}$

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Let $S$ denote the surface of the box $[1, 2] \times [-1, 3] \times [2, 6]$ with the outward pointing orientation, and let \begin{align} \mathbf{G} = (x^2 y + e^y, z^2 - 2x^2, x^2 + 4y^3 + 3z) \end{align} Evaluate $\iint_S \mathbf{G} \cdot d\mathbf{S}$