10 find the divergence of the following vectors a a x

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10) Find the divergence of the following vectors: a) A = x ˆ x + y ˆ y + z ˆ z b) A = xy 2 z 3 ˆ x + ˆ y + ˆ z ( ) c) A = r cos φ ˆ r + z r sin φ ˆ z d) A = r 2 sin θ cos φ ˆ r + ˆ θ + ˆ φ ( ) 11) For the pyramidal shape shown on the right: a) Find the area element d S (both magnitude and direction) for each of the for faces. b) Determine the flux of the position vector A = x ˆ x + y ˆ y + z ˆ z through each of the four faces. 12) Find the curl of the following vectors: a) A = x ˆ x + y ˆ y + z ˆ z b) A = xy 2 z 3 ˆ x + ˆ y + ˆ z ( ) c) A = r cos φ ˆ r + z r sin φ ˆ z d) A = r 2 sin θ cos φ ˆ r + cos θ sin φ r 2 ˆ θ 13) Some of the unit vectors in cylindrical and spherical coordinates change in space, and thus (unlike Cartesian coordinates) are not constant vectors. This means that spatial derivatives of the unit vectors are generally non-zero. Find the divergence and curl of these unit vectors.
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