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The inner product of two column vectors \mathbf a and \mathbf b, \mathbf a \cdot \mathbf b, written in matrix form is \mathbf a^T * \mathbf b.
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Definition of Partial Derivative: Correct Form?
- Thread starter Cyrus
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SUMMARY
The discussion centers on the correct definition of the partial derivative, specifically the notation used to express it. Two forms are presented: one as a row vector, \frac {\partial f}{\partial \bar{x} } = [ \frac {\partial f}{\partial x_1 }, \frac {\partial f}{\partial x_2 }, \frac {\partial f}{\partial x_n }] , and the other as a column vector, \frac {\partial f}{\partial \bar{x} } = \left[\begin{array}{c} \frac {\partial f}{\partial x_1 } \\ \\ \frac {\partial f}{\partial x_2 }\\ \\ \frac {\partial f}{\partial x_n }\end{array}\right]. The consensus leans towards the first definition being correct, as it aligns with standard conventions in mathematics where gradients are represented as row vectors. The context of usage and the transformation rules for displacement vectors and gradients are also discussed, emphasizing the importance of notation in proofs.
- Understanding of vector notation in calculus
- Familiarity with gradient concepts in multivariable calculus
- Knowledge of tensor representation in physics
- Basic principles of linear algebra
- Research the differences between row and column vector representations in linear algebra
- Study the transformation rules for gradients and displacement vectors in tensor calculus
- Explore the implications of notation in mathematical proofs and their impact on interpretation
- Learn about the Jacobian matrix and its applications in multivariable calculus
Mathematicians, physics students, and anyone involved in advanced calculus or differential geometry who needs clarity on the notation and definitions of partial derivatives and gradients.
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