Verify Stokes's Theorem for the following vector field E = xyâg – (x² + 2y²)ây for the contour shown below. Recall Stokes's Theorem: f ë - ai = [[ (v × E) · aš C S y 0-
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- a) Sketch and label the contours z = 0, z = ±1, and ±4 for the function z = f(x, y) = –x² + y?. Plot the gradient vector field F = V f. メ b) Plot the vector field F(x, y) = (y, 0). メConsider the vector field LY f(x, y) = 9e 9 #y a. Find the direction in which the directional derivative of f(x, y), at the point (x, y) = (0,4), has a value of 1. Please input your answer as a column vector. b. Identify one point at which the gradient of the function f(x, y) = 2æ² + 3y² – 6x – 5y is i + j. Give your answer in the form (x, y).Consider the following vector field. F(x, y, 2) = xy?z?i + x?yz?j + x?y?zk %3D (a) Find the curl of the vector field. curl(F) = (b) Find the divergence of the vector field. div(F) =
- A certain radioactive substance has a half-life of 3.6 years. Find how long it takes (in years) for the substance to decompose to 10% of its initial amount.Consider the vector field F = Evaluate Cr(t) = tit³j, 0Consider the vector field F = Evaluate r(t) = Pi+tj, 0Consider the vector field F Evaluate Calculator ww T [F F. dr along the curve C: F(t) = 6 cos(t)i + 6 sin(t)j, 0≤t≤ 4 C Check Answer 3 3 2Given the vector field:Mark the correct statements. i.The value of the vector field can also be just a real number. ii.All vectors in the vector field F(r)= −r/2 point towards the origin. Here r is the position vector of the point (x,y,z). iii.A charged particle always moves along the field line of the electric field if it is not acted upon by other forces. iv.The field lines of a vector field can sometimes be closed curves.Recommended textbooks for youAdvanced Engineering MathematicsAdvanced MathISBN:9780470458365Author:Erwin KreyszigPublisher:Wiley, John & Sons, IncorporatedNumerical Methods for EngineersAdvanced MathISBN:9780073397924Author:Steven C. Chapra Dr., Raymond P. CanalePublisher:McGraw-Hill EducationIntroductory Mathematics for Engineering Applicat…Advanced MathISBN:9781118141809Author:Nathan KlingbeilPublisher:WILEYMathematics For Machine TechnologyAdvanced MathISBN:9781337798310Author:Peterson, John.Publisher:Cengage Learning,Advanced Engineering MathematicsAdvanced MathISBN:9780470458365Author:Erwin KreyszigPublisher:Wiley, John & Sons, IncorporatedNumerical Methods for EngineersAdvanced MathISBN:9780073397924Author:Steven C. Chapra Dr., Raymond P. CanalePublisher:McGraw-Hill EducationIntroductory Mathematics for Engineering Applicat…Advanced MathISBN:9781118141809Author:Nathan KlingbeilPublisher:WILEYMathematics For Machine TechnologyAdvanced MathISBN:9781337798310Author:Peterson, John.Publisher:Cengage Learning,