Verify Formula (1) in the Divergence Theorem by evaluating...

Verify Formula (1) in the Divergence Theorem by evaluating the surface integral and the triple integral. $$ \begin{array}{l}{\mathbf{F}(x, y, z)=x \mathbf{i}+y \mathbf{j}+z \mathbf{k} ; \sigma \text { is the surface of the cube }} \\ {\text { bounded by the planes } x=0, x=1, y=0, y=1, z=0} \\ {z=1}\end{array} $$

Verify Formula (1) in the Divergence Theorem by evaluating the surface integral and the triple integral.
$$
\begin{array}{l}{\mathbf{F}(x, y, z)=x \mathbf{i}+y \mathbf{j}+z \mathbf{k} ; \sigma \text { is the surface of the cube }} \\ {\text { bounded by the planes } x=0, x=1, y=0, y=1, z=0} \\ {z=1}\end{array}
$$

Key Concepts

Divergence

Divergence is a differential operator that measures the magnitude of a source or sink at a given point in a vector field. Formally, it is the scalar result of taking the dot product of the del operator with a vector field, and it quantifies the net rate of flux expansion per unit volume at that point.

Vector Field

A vector field assigns a vector to every point in a region of space. In studies involving the Divergence Theorem, vector fields are typically evaluated for their behavior across volumes and surfaces. The field’s properties, such as its divergence, determine how it contributes to flux across a given boundary.

Triple Integral

A triple integral is used to compute the accumulation of a quantity over a three-dimensional region, such as volume, mass, or in this case, the divergence of a vector field. When applying the Divergence Theorem, the triple integral of the divergence over a volume is set equal to the surface integral of the vector field flux over the boundary of the volume.

Divergence Theorem

The Divergence Theorem, also known as Gauss's Theorem, is a fundamental result in vector calculus that relates the flux of a vector field through a closed surface to the volume integral of its divergence. It essentially states that the total 'outflow' of a vector field from a volume is equal to the integral of the divergence over the volume enclosed by the surface.

Surface Integral

A surface integral is a method used to integrate a function over a surface in three dimensions. In the context of vector fields, it often represents the flux through the surface, which is obtained by taking the dot product of the vector field with the unit normal vector to the surface and integrating this product over the entire surface.

Transcript

00:01 So in this section, we're discussing the divergence theorem.

00:05 And ultimately, what we have is a vector field, f, and it is equal to this vector function of x, y, and z, which is just x, the vector x, y, z, and it's equal to the vector r.

00:27 So then we have this surface that encloses the box.

00:38 It's bounded by the planes.

00:40 X equal to y, equal to z, equal to zero.

00:45 And then x equal to y equal to z equal to one.

00:50 So with that in mind, we now have the first surface, which is x, y, y, and z where zero is less than or equal to zero is less than x which is less than equal to one zero is less than y which is less than or equal to one and z equals zero then we know that the unit normal vector is negative k hat um so it's on the face of the plane z equals zero um so that means that the flux across the face is going to be signified by fee, which is equal to this double integral right here across the first surface, f .n .d .s, which is equal to the double integral of xy, negative k, hat, ds, which we know is just going to be equal to zero.

02:19 So on the face of z equals 1 now, we're going to have something similar.

02:24 It's the second surface right here.

02:29 And that's just going to be x, y, x and y, where 0 is less than or equal to x, which is less than or equal to 1.

02:42 And 0 is less than or equal to y, which is less than or equal to 1.

02:47 So based on that, now what we have is fee being equal to the double integral of the second surface of x, y, 1 times, and this is just 0 ,0, 1...

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