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Proof In parts (a) - (h), prove the property for
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Chapter 15 Solutions
Calculus
- Vector A satisfies the vector transformation law, A'=SA. Show directly that its time derivative dA/dt also satisfies A'=SA and is therefore a vectorarrow_forwardQuestion: Prove that the 2d-curl of a conservative vector field is zero, ( ∇ × ∇ f ) ⋅ k = 0 (here k is unit vector) for any general scalar function f ( x , y ).arrow_forwardIdentities Prove the following identities. Assume φ is a differentiablescalar-valued function and F and G are differentiable vectorfields, all defined on a region of ℝ3. ∇ (F ⋅ G ) = (G ⋅ ∇) F + (F ⋅ ∇)G + G x (∇ x F) + F x (∇ x G)arrow_forward
- Sketching vector fields Sketch the following vector field. F = ⟨1, y⟩arrow_forwardFinding potential functions Determine whether the following vector fields are conservative on the specified region. If so, determine a potential function. Let R* and D* be open regions of ℝ2 and ℝ3, respectively, that do not include the origin.arrow_forwardFinding potential functions Determine whether the following vector fields are conservative on the specified region. If so, determine a potential function. Let R* and D* be open regions of ℝ2 and ℝ3, respectively, that do not include the origin. F = ⟨-y, -x⟩ on ℝ2arrow_forward
- Gravitational potential The potential function for the gravitational force field due to a mass M at the origin acting on a mass m is φ = GMm/ | r | , where r = ⟨x, y, z⟩ is the position vector of the mass m, and G is the gravitational constant.a. Compute the gravitational force field F = -∇φ .b. Show that the field is irrotational; that is, show that ∇ x F = 0.arrow_forwardFinding potential functions Determine whether the following vector fields are conservative on the specified region. If so, determine a potential function. Let R* and D* be open regions of ℝ2 and ℝ3, respectively, that do not include the origin. F = ⟨x3, 2y, -z3⟩ on ℝ3arrow_forwardCurl of a vector field Compute the curl of the following vector field. F = ⟨0, z2 - y2, -yz⟩arrow_forward
- Finding potential functions Determine whether the following vector fields are conservative on the specified region. If so, determine a potential function. Let R* and D* be open regions of ℝ2 and ℝ3, respectively, that do not include the origin. F = ⟨1, -z, y⟩ on ℝ3arrow_forwardRain on a roof Consider the vertical vector field F = ⟨0, 0, -1⟩, correspondingto a constant downward flow. Find the flux in the downward direction acrossthe surface S, which is the plane z = 4 - 2x - y in the first octant.arrow_forwardVector Fields and Divergence.arrow_forward
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