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Consider the following vector function: A = (6xy) x + (3x²)ŷ + (4z)2 a) Calculate the divergence and the curl of this vector function. b) Calculate the path integral of this vector function from the origin to the point (1,1,1) using two different paths: 1) going in the direction from (0,0,0) to (1,0,0), then the y direction from (1,0,0) to (1,1,0), then the 2 direction from (1,1,0) to (1,1,1) and 2) going in a straight line from (0,0,0) to (1,1,1). | =) Are the results of parts a) and b) consistent with each other? Explain why or why not.

Consider the following vector function: A = (6xy) x + (3x²)ŷ + (4z)2 a) Calculate the divergence and the curl of this vector function. b) Calculate the path integral of this vector function from the origin to the point (1,1,1) using two different paths: 1) going in the direction from (0,0,0) to (1,0,0), then the y direction from (1,0,0) to (1,1,0), then the 2 direction from (1,1,0) to (1,1,1) and 2) going in a straight line from (0,0,0) to (1,1,1). | =) Are the results of parts a) and b) consistent with each other? Explain why or why not.

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Consider the following vector function: \(\mathbf{A} = (6xy)\hat{x} + (3x^2)\hat{y} + (4z)\hat{z}\) a) Calculate the divergence and the curl of this vector function. [Blank space] b) Calculate the path integral of this vector function from the origin to the point \((1,1,1)\) using two different paths: 1) Going in the \(\hat{x}\) direction from \((0,0,0)\) to \((1,0,0)\), then the \(\hat{y}\) direction from \((1,0,0)\) to \((1,1,0)\), then the \(\hat{z}\) direction from \((1,1,0)\) to \((1,1,1)\) 2) Going in a straight line from \((0,0,0)\) to \((1,1,1)\). [Blank space] c) Are the results of parts a) and b) consistent with each other? Explain why or why not. [Blank space]
Transcribed Image Text:Consider the following vector function: \(\mathbf{A} = (6xy)\hat{x} + (3x^2)\hat{y} + (4z)\hat{z}\) a) Calculate the divergence and the curl of this vector function. [Blank space] b) Calculate the path integral of this vector function from the origin to the point \((1,1,1)\) using two different paths: 1) Going in the \(\hat{x}\) direction from \((0,0,0)\) to \((1,0,0)\), then the \(\hat{y}\) direction from \((1,0,0)\) to \((1,1,0)\), then the \(\hat{z}\) direction from \((1,1,0)\) to \((1,1,1)\) 2) Going in a straight line from \((0,0,0)\) to \((1,1,1)\). [Blank space] c) Are the results of parts a) and b) consistent with each other? Explain why or why not. [Blank space]
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