1-8 SCALAR FIELDS IN THE PLANE Let the temperature T' in a body be independent of z, so that it is given by a scalar function T = T(x, t). Identify the isotherms T(x, y) = const. Sketch some of them. 1. T = x² - y2 2. T = xy 3. T = 3x - 4y 4. T = arctan (y/x) 5. T = y/(x² + y²) 6. T = x/(x² + y²) 7. T = 9x² + 4y² 8. CAS PROJECT. Scalar Fields in the Plane. Sketch or graph isotherms of the following fields and describe what they look like. (b) x²y - y³/3 (a) x² - 4x - y² (c) cos x sinh y (e) e sin y (d) sin x sinh y (f) 2x cos 2y (h) x² - 2x - y2 (g) x4 - 6x²y2 + y4 SCALAR FIELDS IN SPACE 9-14 What kind of surfaces are the level surfaces f(x, y, z) = const? 10. f = 9(x² + y²) + z² 9. f= 4x - 3y + 2z 11. f = 5x² + 2y² 13. f=z-(x² + y²) 12. f = z √x² + y2 14. f = x - y²