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Prove each identity, assuming that S and E satisfy the conditions of the Divergence Theorem and the scalar functions and components of the
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Multivariable Calculus
- Let f1(x)=3x and f2(x)=|x|. Graph both functions on the interval 2x2. Show that these functions are linearly dependent in the vector space C[0,1], but linearly independent in C[1,1].arrow_forwardFind the gradient vector for the vector field.arrow_forward42. Derivatives of triple scalar products a. Show that if u, v, and w are differentiable vector functions of t, then du v X w + u• dt dv X w + u•v X dt dw (u•v X w) dt dt b. Show that d'r dr? dr dr d'r dt dt r. dt dr? (Hint: Differentiate on the left and look for vectors whose products are zero.)arrow_forward
- Please give a clear solution.arrow_forward4. Assume A is a constant vector field and R = xˆi + yˆj + z ˆk. (a) Simplify the expression (R · ∇R ) · A+ ∇ · ∇ × ∇ × ((A× R ) × (R × A))arrow_forwardUse the equation giving the flux of the vector field across the curve to calculate the flux of x + 1 y lã (x + 1)² + y²' (x + 1)² + y² F(x, y) = across C, the segment 7 ≤ y ≤ 9 along the y-axis, oriented upwards. (Use symbolic notation and fractions where needed.) I F. dr =arrow_forward
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