Verify that the Divergence Theorem is true for the vector...

Key Concepts

Divergence Theorem

The Divergence Theorem is a fundamental result in vector calculus that relates the flux of a vector field across a closed surface to the integral of the divergence of the field over the volume enclosed by that surface. It states that the total quantity of a vector field 'exiting' a region can be calculated by integrating the divergence of the field throughout the region, thus linking surface integrals to volume integrals.

Divergence of a Vector Field

The divergence of a vector field is an operator that measures the rate at which the vector field spreads out from a given point. Algebraically, it is computed as the sum of the partial derivatives of each vector component with respect to its corresponding coordinate variable. This scalar quantity is crucial in applying the Divergence Theorem as it forms the integrand in the volume integral.

Volume Integrals

Volume integrals are used to aggregate quantities over a three-dimensional region, which, in the context of the Divergence Theorem, are used to integrate the divergence of a vector field. They provide a method to calculate properties like mass, charge, or flux density distributed throughout a volume.

Surface Integrals

Surface integrals extend the concept of an integral to functions defined on surfaces, typically representing quantities like flux. In the application of the Divergence Theorem, surface integrals are used to compute the net flow of a vector field through the boundary of a volume, summing contributions from each infinitesimal surface element.

Transcript

00:01 So we're given here f of xyz, that equals 3x i cap plus xy cap plus xy gap plus 2x z k gap.

00:13 This we have.

00:14 And e, which is the cube bounded by the planes, we have x is equal to 0, x is equal to 1, y equal to 0, y is equal to 1, z is equal to 1, z is equal to 1, and 0 is 8 equal to 1.

00:31 So, this we have here, let me just verify this.

00:40 Yeah, it is verified.

00:42 So now we just continue.

00:44 The solutions will have p is equal to 3x, q as equal to xy, and r will be 2xc.

00:54 We need to solve now the integrals, that is double integral, f.

01:00 Dot d s, right? we need to solve that over s and triple integral over e divergence of f dv.

01:10 All right.

01:12 That's going to solve here.

01:18 All right.

01:19 Now we have here divergence of f that equals curly p over curly x plus curly q over curly y plus curly r over curly z.

01:32 So we do that we will get 3 plus x plus 2x.

01:37 Three times x plus one right we calculate now the first integral the triple integral e divergence f dv that is given as 0 to 1 0 2 1 0 2 1 3 times x plus 1 d z ty d x right so this we have here so we keep on solving this we get 3 times 0 2 1 0 to 1 will give 3 times x plus 1 c that is from 0 to 1 d by d x right now we keep on solving this we get 3 times 0 2 1 0 2 1 x plus 1 d by d x we solve double integral we will get it as 9 over 2 this is coming up to be 9 over 2 here next we need to find out f dot d s over double integral over double integral s f dot d s that is given as we have double integral s1 f dot ds plus and so on so we'll give here s6 f dot d s so we have surface s one that will be x is equal to one plus that zero less than equal to y less than equal to 1 0 less than equal to z less than equal to 1 and the normal vector n1 will be i cap so we will have double integral f dot ds f dot ds right over s 1 so is giving as s 1 f dot n1 d s that is given as double integral s 1 3 i cap plus y j cap plus 2 z k gap dot i cap d s that will give double integral s 1 3 d s that is given as 3 right then we have surface s 2 s 2 is given as y as equal to 1 such that 0 less than equal to x less than equal to 1 0 less than equal to z less than equal to 1 and normal vector n1 is equal to j -calf.

04:31 So we have double integral as to f dot ds.

04:36 That equals double integral as to f dot n2 vector ds.

04:45 That would give double integral as to 3x i cap plus x jcap plus 2x zkkap dot jds.

04:58 We solve this we will get in s 1 over 2.

05:04 Then we have surface s3.

05:13 S3 is z is equal to 1 plus that 0 less than equal to x less than equal to 1, 0 less than equal to y, less than equal to 1.

05:22 And normal vector in 3, that equals k vector...

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