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Vsevolod [243]
3 years ago
6

Determine whether or not f is a gradient vector field. if it is, find a function φ such that f = ∇φ. (if the vector field is not

a gradient field, enter dne.) f(x, y) = (2x − 6y) i + (−6x + 10y − 7) j
Mathematics
1 answer:
MArishka [77]3 years ago
4 0
\mathbf f(x,y)=(2x-6y)\,\mathbf i+(-6x+10y-7)\,\mathbf j

For \mathbf f to be the gradient of some scalar function \varphi(x,y), we need to find \varphi that satisfies

\dfrac{\partial \varphi}{\partial x}=2x-6y
\dfrac{\partial \varphi}{\partial y}=-6x+10y-7

From the first PDE, it follows that (upon integrating with respect to x)

\varphi(x,y)=x^2-6xy+g(y)

Differentiating with respect to y yields

\dfrac{\partial\varphi}{\partial y}=-6x+10y-7=-6x+\dfrac{\mathrm dg}{\mathrm dy}
\implies\dfrac{\mathrm dg}{\mathrm dy}=10y-7\implies g(y)=5y^2-7y+C

\implies\varphi(x,y)=x^2-6xy+5y^2-7y+C
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2 years ago
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Lostsunrise [7]
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3 years ago
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The diameter of the attic is 18ft and its height is 18ft. What's is the volume of the space in the attic
Svet_ta [14]
I attached an image with the complete question.

Answer:
volume of attic = 1526.04 ft³

Explanation:
Volume of cone can be calculated using the following rule:
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We are given that:
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Substitute in the equation to get the volume as follows:
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Hope this helps :)

5 0
3 years ago
Let f(x) = -2x + 4 and g(x) = 5x - 7. Find f(x) - g(x).
balandron [24]

Answer:

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4 0
3 years ago
What is the answer please help
Alekssandra [29.7K]

Answer:

x^2 + 3x -1

Step-by-step explanation:

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Combine like terms

           = x^2 + 3x -1

8 0
2 years ago
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