SUMMARY
The directional derivative of the function \(\varphi = x^2 + \sin y - xz\) in the direction of the vector \( \mathbf{u} = \mathbf{i} + 2\mathbf{j} - 2\mathbf{k} \) at the point \((1, \frac{\pi}{2}, -3)\) is calculated using the normalized direction vector \(\hat{u} = \frac{1}{3}\mathbf{i} + \frac{2}{3}\mathbf{j} - \frac{2}{3}\mathbf{k}\) and the gradient \(\nabla\varphi = (2x - z)\mathbf{i} + \cos y \mathbf{j} - x\mathbf{k}\). At the specified point, the gradient evaluates to \((5, 0, -1)\). The final directional derivative is \(\frac{7}{3}\).
PREREQUISITES
- Understanding of directional derivatives in multivariable calculus
- Familiarity with gradient vectors and their properties
- Knowledge of vector normalization techniques
- Basic proficiency in trigonometric functions, specifically sine and cosine
NEXT STEPS
- Study the concept of gradient vectors in multivariable calculus
- Learn about the properties and applications of directional derivatives
- Explore vector normalization methods in detail
- Investigate the implications of trigonometric functions in calculus
USEFUL FOR
Students and professionals in mathematics, particularly those studying multivariable calculus, as well as educators looking to enhance their understanding of directional derivatives and gradient calculations.