Find the Gradient Field of the Function $$f ( x , y , z ) = \left( x ^ { 8 } y ^ { 2 } - y ^ { 9 } z ^ { 3 } \right) e ^ { - z ^ { 2 } }$$

question 190

Multiple Choice

Find the gradient field of the function.
- $$f ( x , y , z ) = \left( x ^ { 8 } y ^ { 2 } - y ^ { 9 } z ^ { 3 } \right) e ^ { - z ^ { 2 } }$$

• A)\({\nabla} f= 8 x ^ { 7 } y ^ { 2 } e ^ { - z ^ { 2 } } \mathbf { i } + \left( 2 x ^ { 8 } y - 9 y ^ { 8 } z ^ { 3 } \right) e ^ { - z ^ { 2 } } \mathbf { j } + y ^ { 2 } z e ^ { - z ^ { 2 } } \left[ y ^ { 7 } z \left( 2 z ^ { 2 } - 3 \right) - 2 x ^ { 8 } \right] \mathbf { k }\)
• B) \({\nabla} f = 8 \mathrm { x } ^ { 7 } \mathbf { i } + \left( 9 \mathrm { y } ^ { 8 } - 2 \mathrm { y } \right) \mathbf { j } + 2 \mathrm { z } ^ { 4 } \mathrm { e } ^ { - \mathrm { z } ^ { 2 } } \mathbf { k }\)
• C) \({\nabla} f = 8 \mathrm { x } ^ { 7 } \mathrm { y } ^ { 2 } \mathrm { e } ^ { - \mathrm { z } ^ { 2 } } \mathrm { i } + \left( 2 \mathrm { x } ^ { 8 } \mathrm { y } - 9 \mathrm { y } ^ { 8 } \mathrm { z } ^ { 3 } \right) \mathrm { e } ^ { - \mathrm { z } ^ { 2 } } \mathrm { j } + \left( 3 - 2 \mathrm { z } ^ { 2 } \right) \mathrm { y } ^ { 9 } \mathrm { z } ^ { 2 } \mathrm { e } ^ { - \mathrm { z } ^ { 2 } } \mathbf { k }\)
• D) \(\nabla f = 8 x ^ { 7 } y ^ { 2 } \mathbf { i } + \left( 9 y ^ { 8 } z ^ { 3 } - 2 x ^ { 8 } y \right) \mathbf { j } + \left( 2 z ^ { 2 } - 3 \right) y { } ^ { 9 } { } ^ { 2 } \mathbf { k }\)

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