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- Determine if the scalar vector field is Lipschitz Continous on the indicated domain. f(t,x) = SiN (Ex), _√2 = R, I= R XFind the directional derivative of the scalar field þ= x¹y + 4xz at the point (2, -2, 4) along the direction vector (2, -1, -2):determine if the given vector field f is conservative and hence find the potential function for the vector field if it is conservative F=(6x^2-2xy^2+y/2√x) i-(2x^2y-4-√x) j.
- A net is dipped in a river. Determine the flow rate of water across the net if the velocity vector field for the river is given by v=(x-y,z+y+7,z2) and the net is decribed by the equation y=1-x2-z2, y20, and oriented in the positive y- direction. (Use symbolic notation and fractions where needed.)Find a scalar field, 0, whose gradient is the irrotational vector V = (x² - yz, y² — xz, ² - xy) In other words, find such that Vo=V₁ Note: • Remember that your tutorial has Hints in it, at the end of each chapter. If you're stuck, they are often quite useful! • If the answer is a scalar, you can just type it in the box (using the Calcpad if you like, or using / for fractions, ^ for exponents, shift and - for subscripts, etc.) For multiplication, you can either leave a space, or use *. So x* x = xx = x². Note that this is not the same as x without a space; that will get read as an entirely different variable! • If you need to enter a vector, enter an ordered list of components, so for A you can enter either A = (Az. Ay, A₂) = {Az, Ay, A₂}. Note that the system isn't great with multiplying through by overall factors, so it's better not to write e.g. o= (2A, 2A, 2A₂). + X .. 15 calc Opera Functi Symb RelatiCompute the vector assigned to the points P = (1, 1) and Q = (-3, -3) by the vector field F = (-y, x). F(1, 1) = F(-3, -3) -
- Compute and sketch the vector assigned to the points P = (1, 2)and Q = (−1, −1) by the vector field F = (x2, x)A solid material has thermal conductivity K in kilowatts per meter-kelvin and temperature given at each point by w(x, y, z) = 15 - 4(x² + y² + z²) °C. Use the fact that heat flow is given by the vector field F = -KVw and the rate of heat flow across a surface S within the solid is given by -K , VwdS. Find the rate of heat flow out of a sphere of radius 1 (centered at the origin) inside a large cube of copper (K = 400 kW/(m · K)). (Use symbolic notation and fractions where needed.) -K Incorrect D VwdS= 4.104 kWLet F and G be vector fields with differentiable components. Express curl (F x G) in term of div and dot products.