The gradient?
The gradient of a function of two variables, ∇ f ( x , y ) , is always perpendicular to the graph of the function.
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1 solution
Yeah....level curves........or as I prefer to use......perpendicular to the contour lines..........
The statement is not even false but meaningless, since the graph resides in R 3 while the gradient resides in R 2 .
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Yeah as my professor puts it "The gradient lives in the domain"
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Exactly! So maybe you should change the answer to a third option, "the statement is meaningless."
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@Otto Bretscher – I think that this question may help in clearing the concepts of some people who are confused regarding gradients..........
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@Aaghaz Mahajan – It does not really help to say that the statement is false. The main point is that the gradient and the graph live in different spaces.
As a Swiss, I like to explain this stuff in terms of landscapes ("mountains and valleys") and maps. The gradient lives on the map, and it is meaningless (not even false) to say that it is perpendicular to the landscape.
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@Otto Bretscher – Well,Sir....that is obvious....but what I was saying is that maybe some people are not clear with the definition of gradient itself........let alone where it "lives" in........But, that is my opinion.... :) Hummus can always change the options to whatever he seems is correct..............
@Otto Bretscher – The statement itself is false regardless if it is deemed meaningless given the context that the gradient lives in the domain. The 2 dimensional gradient here by definition has a z component of 0.
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@Hamza A – No! In maths we don't do that; R 2 is not a subspace of R 3 , although this identification is sometimes made in introductory physics courses, unfortunately. In maths, ( x , y ) is not identified with ( x , y , 0 ) ; as we keep saying: they live in different spaces.
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@Otto Bretscher – I see. Can you report it and say that in the report? Thank you.
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@Hamza A – Let me do that; it will be interesting to see what the experts from Brilliant think. Another point: The term "three-dimensional function" is unusual at best; the common term is "function of two variables."
An important distinction is to be made here. The gradient is perpendicular to the level curves rather than the graph of a function