2. A vector field is defined in a Cartesian coordinate system by 22 3 vec(A) =...

 

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2. A vector field is defined in a Cartesian coordinate system by 22 3 vec(A) = 3x²y zhat(1) + 2x³yzhat(1) + x³y²hat(k) (a) Show that for this field vec(grad) x vec(A) = vec(0). (b) The previous result implies that vec(A) = vec(grad), where p is a scalar function of position. Determine (x, y, z) subject to the condition that p = 0 at the origin. (c) The curve C is defined parametrically by x = t², y = 2t,z= t³ from t = 0 to t = 1. i. Show that, along C, 4 = 4t¹¹. ii. Evaluate fødvec(r) where vec(r) is the position vector of a point on C. 2. A vector field is defined in a Cartesian coordinate system by 22 3 vec(A) = 3x²y zhat(1) + 2x³yzhat(1) + x³y²hat(k) (a) Show that for this field vec(grad) x vec(A) = vec(0). (b) The previous result implies that vec(A) = vec(grad), where p is a scalar function of position. Determine (x, y, z) subject to the condition that p = 0 at the origin. (c) The curve C is defined parametrically by x = t², y = 2t,z= t³ from t = 0 to t = 1. i. Show that, along C, 4 = 4t¹¹. ii. Evaluate fødvec(r) where vec(r) is the position vector of a point on C.

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a To show that vecgrad x vecA vec0 we need to compute the cross product and demonstrate that it results in the zero vector The gradient operator in Ca... View the full answer