Let S1 be the level surface given by the equation f(x,y,z) =k1and letS2be the level surface...

Let S1 be the level surface given by the equation f(x,y,z) =k1and letS2be the level surface given by the equation g(x,y,z) =k2. Suppose that C is the curve of intersection of the surfacesS1andS2andassume that P is a point on C. show that any tangent vector to C at P must be parallel to∇f(P)×∇g(P)

Expert Solution

Colby Messinger answered 2 years ago

The curried version of let f (x,y,z) = (x,(y,z)) is let f (x,(y,z)) = (x,(y,z)) Just...

The curried version of let f (x,y,z) = (x,(y,z)) is let f (x,(y,z)) = (x,(y,z)) Just f (because f is already curried) let f x y z = (x,(y,z)) let f x y z = x (y z)

5) Let the function f : ℝ3 → ℝ3 be given by f(x, y, z) =...

5) Let the function f : ℝ3 → ℝ3 be given by f(x, y, z) = (2x + 2y, 2y + 2z, z + x). a) Prove that f is one to one and onto b) Find the inverse of f, i.e., f−1.

5. Let X, Y and Z be sets. Let f : X ! Y and g...

5. Let X, Y and Z be sets. Let f : X ! Y and g : Y ! Z functions. (a) (3 Pts.) Show that if g f is an injective function, then f is an injective function. (b) (2 Pts.) Find examples of sets X, Y and Z and functions f : X ! Y and g : Y ! Z such that g f is injective but g is not injective. (c) (3 Pts.) Show that if...

Let f(x,y) be a scalar function, and let F(x,y,z) be a vector field. Only one of...

Let f(x,y) be a scalar function, and let F(x,y,z) be a vector field. Only one of the following expressions is meaningful. Which one? a) grad f x div F b) div(curl(grad f)) c) div(div F) d) curl(div(grad f)) e) grad(curl F)

Let X, Y ⊂ Z and x, y ∈ Z Let A = (X\{x}) ∪ {x}....

Let X, Y ⊂ Z and x, y ∈ Z Let A = (X\{x}) ∪ {x}. a) Prove or disprove: A ⊆ X b) Prove or disprove: X ⊆ A c) Prove or disprove: P(X ∪ Y ) ⊆ P(X) ∪ P(Y ) ∪ P(X ∩ Y ) d) Prove or disprove: P(X) ∪ P(Y ) ∪ P(X ∩ Y ) ⊆ P(X ∪ Y )

Describe the level surfaces for the 3-variable function: f(x,y,z) = z/(x-y)

Let f: X-->Y and g: Y-->Z be arbitrary maps of sets (a) Show that if f...

Let f: X-->Y and g: Y-->Z be arbitrary maps of sets (a) Show that if f and g are injective then so is the composition g o f (b) Show that if f and g are surjective then so is the composition g o f (c) Show that if f and g are bijective then so is the composition g o f and (g o f)^-1 = g ^ -1 o f ^ -1 (d) Show that f: X-->Y is...

Prove Proposition 6.10 (Let f : X → Y and g : Y → Z be...

Prove Proposition 6.10 (Let f : X → Y and g : Y → Z be one to one and onto functions. Then g ◦ f : X → Z is one to one and onto; and (g ◦ f)−1 = f−1 ◦ g−1 ).

3. Let F : X → Y and G: Y → Z be functions. i. If...

3. Let F : X → Y and G: Y → Z be functions. i. If G ◦ F is injective, then F is injective. ii. If G ◦ F is surjective, then G is surjective. iii. If G ◦ F is constant, then F is constant or G is constant. iv. If F is constant or G is constant, then G ◦ F is constant.

Let f : X → Y and g : Y → Z be functions. We can...

Let f : X → Y and g : Y → Z be functions. We can deﬁne the composition of f and g to be the function g◦ f : X → Z for which the image of each x ∈ X is g(f(x)). That is, plug x into f, then plug theresultinto g (justlikecompositioninalgebraandcalculus). (a) If f and g arebothinjective,must g◦ f beinjective? Explain. (b) If f and g arebothsurjective,must g◦ f besurjective? Explain. (c) Suppose g◦ f isinjective....

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