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Multivariable Calculus
- ProofProve in full detail that the set {(x,2x):xisarealnumber}, with the standard operations in R2, is a vector space.arrow_forwardCAPSTONE (a) Explain how to determine whether a function defines an inner product. (b) Let u and v be vectors in an inner product space V, such that v0. Explain how to find the orthogonal projection of u onto v.arrow_forwardCalculus Determine whether the set S={fC[0,1]:01f(x)dx=0} is a subspace of C[0,1]. Prove your answer.arrow_forward
- Identities 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. ∇ x (∇ x F) = ∇(∇ ⋅ F) - (∇ ⋅ ∇)Farrow_forwardDerivative of vector functions Compute the derivative of the followingfunctions.a. r(t) = ⟨t3, 3t2, t3/6⟩ b. r(t) = e-t i + 10√t j + 2 cos 3t karrow_forwardDerivatives of vector-valued functions Differentiate the following function. r(t) = tan t i + sec t j + cos2 t karrow_forward
- Prove the property. In each case, assume r, u, and v are differentiable vector-valued functions of t in space, w is a differentiable real-valued function of t, and c is a scalar. d/dt [r(t) × u(t)] = r(t) × u′(t) + r′(t) × u(t)arrow_forwardSketching vector fields Sketch the following vector field. F = ⟨x, 0⟩arrow_forwardWhat is the graph of r(t)=2 costi + sintj? Calculus 3 Vector valued functions and space curves.arrow_forward
- Splitting a vector field Express the vector field F = ⟨xy, 0, 0⟩in the form V + W, where ∇ ⋅ V = 0 and ∇ x W = 0.arrow_forwardProperties of div and curl Prove the following properties of thedivergence and curl. Assume F and G are differentiable vectorfields and c is a real number.a. ∇ ⋅ (F + G) = ∇ ⋅ F + ∇ ⋅ Gb. ∇ x (F + G) = (∇ x F) + (∇ x G)c. ∇ ⋅ (cF) = c(∇ ⋅ F)d. ∇ x (cF) = c(∇ ⋅ F)arrow_forwardUsing a Function (a) find the gradient of the function at P, (b) find a unit normal vector to the level curve f (x, y) = c at P, (c) find the tangent line to the level curve f (x, y) = c at P, and (d) sketch the level curve, the unit normal vector, and the tangent line in the xy-plane. f(x, y) = 9x2 + 4y2, c = 40, P(2, −1)arrow_forward
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