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Dr. Hugh Blanton ENTC 3331

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Gradient, Divergence and Curl: the Basics

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Dr. Blanton - ENTC Gradient, Divergence, & Curl 3 We first consider the position vector, l : where x, y, and z are rectangular unit vectors.

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Dr. Blanton - ENTC Gradient, Divergence, & Curl 4 Since the unit vectors for rectangular coordinates are constants, we have for d l :

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Dr. Blanton - ENTC Gradient, Divergence, & Curl 5 The operator, del: is defined to be (in rectangular coordinates) as: This operator operates as a vector.

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Dr. Blanton - ENTC Gradient, Divergence, & Curl 6 Gradient If the del operator, operates on a scalar function, f(x,y,z), we get the gradient:

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Dr. Blanton - ENTC Gradient, Divergence, & Curl 7 We can interpret this gradient as a vector with the magnitude and direction of the maximum change of the function in space. We can relate the gradient to the differential change in the function:

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Dr. Blanton - ENTC Gradient, Divergence, & Curl 8 Directional derivatives: aT dl dT l ˆ

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Dr. Blanton - ENTC Gradient, Divergence, & Curl 9 Since the del operator should be treated as a vector, there are two ways for a vector to multiply another vector: dot product and cross product.

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Dr. Blanton - ENTC Gradient, Divergence, & Curl 10 Divergence We first consider the dot product: The divergence of a vector is defined to be: This will not necessarily be true for other unit vectors in other coordinate systems.

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Dr. Blanton - ENTC Gradient, Divergence, & Curl 11 To get some idea of what the divergence of a vector is, we consider Gauss' theorem (sometimes called the divergence theorem).

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Dr. Blanton - ENTC Gradient, Divergence, & Curl 12 Gauss' Theorem (Gau ’s Theorem We start with: Surface Areas

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Dr. Blanton - ENTC Gradient, Divergence, & Curl 13 We can see that each term as written in the last expression gives the value of the change in vector A that cuts perpendicular through the surface.

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Dr. Blanton - ENTC Gradient, Divergence, & Curl 14 For instance, consider the first term: The first part: gives the change in the x-component of A

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Dr. Blanton - ENTC Gradient, Divergence, & Curl 15 The second part, gives the yz surface (or x component of the surface, S x ) where we define the direction of the surface vector as that direction that is perpendicular to its surface.

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Dr. Blanton - ENTC Gradient, Divergence, & Curl 16 The other two terms give the change in the component of A that is perpendicular to the xz (S y ) and xy (S z ) surfaces.

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Dr. Blanton - ENTC Gradient, Divergence, & Curl 17 We thus can write: where the vector S is the surface area vector.

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Dr. Blanton - ENTC Gradient, Divergence, & Curl 18 Thus we see that the volume integral of the divergence of vector A is equal to the net amount of A that cuts through (or diverges from) the closed surface that surrounds the volume over which the volume integral is taken. Hence the name divergence for

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Dr. Blanton - ENTC Gradient, Divergence, & Curl 19 So what? Divergence literally means to get farther apart from a line of path, or To turn or branch away from.

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Dr. Blanton - ENTC Gradient, Divergence, & Curl 20 Consider the velocity vector of a cyclist not diverted by any thoughts or obstacles: Goes straight ahead at constant velocity. (degree of) divergence 0

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Dr. Blanton - ENTC Gradient, Divergence, & Curl 21 Now suppose they turn with a constant velocity diverges from original direction (degree of) divergence 0

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Dr. Blanton - ENTC Gradient, Divergence, & Curl 22 Now suppose they turn and speed up. diverges from original direction (degree of) divergence >> 0

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Dr. Blanton - ENTC Gradient, Divergence, & Curl 23 Current of water No divergence from original direction (degree of) divergence = 0

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Dr. Blanton - ENTC Gradient, Divergence, & Curl 24 Current of water Divergence from original direction (degree of) divergence ≠ 0

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Dr. Blanton - ENTC Gradient, Divergence, & Curl 25 E-field between two plates of a capacitor. Divergenceless +

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Dr. Blanton - ENTC Gradient, Divergence, & Curl 26 -field inside a solenoid is homogeneous and divergenceless. I divergenceless solenoidal

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Dr. Blanton - ENTC Gradient, Divergence, & Curl 27

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CURL

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Dr. Blanton - ENTC Gradient, Divergence, & Curl 29 Two types of vector fields exists: + + Electrostatic Field where the field lines are open and there is circulation of the field flux. Magnetic Field where the field lines are closed and there is circulation of the field flux. circulation (rotation) = 0 circulation (rotation) 0

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Dr. Blanton - ENTC Gradient, Divergence, & Curl 30 The mathematical concept of circulation involves the curl operator. The curl acts on a vector and generates a vector.

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Dr. Blanton - ENTC Gradient, Divergence, & Curl 31 In Cartesian coordinate system:

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Dr. Blanton - ENTC Gradient, Divergence, & Curl 32 Example

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Dr. Blanton - ENTC Gradient, Divergence, & Curl 33 Important identities: for any scalar function V.

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Dr. Blanton - ENTC Gradient, Divergence, & Curl 34 Stoke’s Theorem General mathematical theorem of Vector Analysis: Any surface Closed boundary of that surface.

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Dr. Blanton - ENTC Gradient, Divergence, & Curl 35 Given a vector field Verify Stoke’s theorem for a segment of a cylindrical surface defined by:

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Dr. Blanton - ENTC Gradient, Divergence, & Curl 36 y x z

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Dr. Blanton - ENTC Gradient, Divergence, & Curl 41 Note that has only one component:

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Dr. Blanton - ENTC Gradient, Divergence, & Curl 42 The integral of over the specified surface S is

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Dr. Blanton - ENTC Gradient, Divergence, & Curl 43

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Dr. Blanton - ENTC Gradient, Divergence, & Curl 44 y x z a b c d

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Dr. Blanton - ENTC Gradient, Divergence, & Curl 45 The surface S is bounded by contour C = abcd. The direction of C is chosen so that it is compatible with the surface normal by the right hand rule.

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Dr. Blanton - ENTC Gradient, Divergence, & Curl 48

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Dr. Blanton - ENTC Gradient, Divergence, & Curl 49 Curl

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Dr. Blanton - ENTC Gradient, Divergence, & Curl 50

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Dr. Blanton - ENTC Gradient, Divergence, & Curl 51 curl or rot place paddle wheel in a river no rotation at the center rotation at the edges

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Dr. Blanton - ENTC Gradient, Divergence, & Curl 52 the vector u n is out of the screen right hand rule s is surface enclosed within loop closed line integral

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Dr. Blanton - ENTC Gradient, Divergence, & Curl 53 Electric Field Lines Rules for Field Lines 1.Electric field lines point to negative charges 2.Electric field lines extend away from positive charges 3.Equipotential (same voltage) lines are perpendicular to a line tangent of the electric field lines

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